###################################################################### # # Maple formulae of some icosahedral solutions (from math.AG/0406281) # Philip Boalch, May 2005 # ###################################################################### # # We will list the solution curves F(y,t)=0 defining y(t) implicitly. # (All have been checked to be solutions algebraically.) # # If this curve is genus zero we will also give a rational parameterisation: # y(s), t(s) such that F(y(s),t(s))=0 for all s. # # If the curve is genus one we will give an elliptic parameterisation: # y(s,u), t(s,u) such that F(y(s,u),t(s,u))=0 for all (s,u) # on the elliptic curve u^2=u2(s), where u2 is the given cubic polynomial. # # The rational parameterisations were generally found by the command: # with(algcurves); # `algcurves/iss94`(F,y,t,s,[Y0, T0, Z0]); # where [Y0, T0, Z0] is a nonsingular rational point of the curve F=0, # written in homogeneous coordinates. # # The elliptic parameterisations were generally found by first using a # symmetry of the curve to reduce to the genus zero case. # ###################################################################### ################################################################# # icosahedral soln 20 # th:=[2/5, 1/3, 1/5, 2/3]; F:=-27*y^5+125*y^5*t-98*t^2+54*t^3-54*t^4+355*y*t^2+135*y*t^3+10*y^2*t-990*y^2*t^2+240*y^3*t+740*y^3*t^2-240*y^4*t-250*y^4*t^2; y:=-2*(2/s-5)*(7/s^2+1/s+1)/s/(5/s+1)/(10/s^2-5/s+4); t:= 27/s^3*(2/s-5)^2/(5/s+1)/(10/s^2-5/s+4)^2; ################################################################ #####simple maple code to check parameterised rational solutions: # first set up the theta values th[1],th[2],th[3],th[4], and y(s),t(s) yp:=diff(y,s);tp:=diff(t,s);dydt:=yp/tp; ypp:=diff(yp,s);tpp:=diff(tp,s);dypdt:=ypp/tp;dtpdt:=tpp/tp; d2ydt2:=dypdt/tp-yp*dtpdt/tp^2; f0:=th[1]; f2:=th[2]; f1:=th[3]; f9:=th[4]; al:=(f9-1)^2/2;be:=-f0^2/2;ga:=f1^2/2;de:=(1-f2^2)/2; f:=-d2ydt2 +(1/y+1/(y-1)+1/(y-t))*(dydt)^2/2 -(1/t+1/(t-1)+1/(y-t))*dydt +((y*(y-1)*(y-t))/(t^2*((t-1)^2)))* (al+be*t/(y^2)+ga*(t-1)/((y-1)^2)+de*t*(t-1)/((y-t)^2)); simplify(f); ###########################################################################3 #icosahedral soln 22 th:=[1/5, 1/5, 2/5, 1/3]; F:= 3456*t^6-189*t^5+(3456*t^6-23436*t^5+378*t^4)*y+(-21870*t^5+70875*t^4)*y^2+ (-7180*t^5+67280*t^4-125440*t^3)*y^3+(-3645*t^5+31530*t^4-113280*t^3+134400*t^2)*y^4+ (11502*t^4-54912*t^3+109824*t^2-86016*t)*y^5+ (3267*t^4-20736*t^3+49408*t^2-57344*t+28672)*y^6; y:=-54*(s-7)*s/(-35-20*s^2+s^4)/(s+1)/(s-4); t:=432*s/(5+s)/(s-4)^2/(s+1)^3; ###########################################################################3 # icosahedral soln 23: th:=[2/5, 2/5, 1/5, 2/3]; F:=-32*t^4+81*t^3+(48*t^4-180*t^3-162*t^2)*y+(150*t^3+585*t^2)*y^2+ (-100*t^3-880*t^2)*y^3+(-15*t^3+750*t^2)*y^4+(-246*t^2-48*t)*y^5+(49*t^2-16*t+16)*y^6; y:=18*s*(-3+s)/(s-4)/(s+1)/(s^2+5); t:=432*s/(5+s)/(s-4)^2/(s+1)^3; ############### #icosahedral soln 24 th:=[1/2, 2/5, 1/5, 4/5]; F:= 343*t^2-686*t^3+527*t^4-184*t^5+48*t^6+(-1372*t^2+2008*t^3-1004*t^4-16*t^5)*y+ (2722*t^2-2868*t^3+1514*t^4-24*t^5)*y^2+(16*t-3404*t^2+1352*t^3-652*t^4)*y^3+ (-40*t+3815*t^2-430*t^3+15*t^4)*y^4+(-288*t-2704*t^2+304*t^3)*y^5+ (472*t+872*t^2)*y^6+(-192*t-192*t^2)*y^7+16*(-1+4*t)*y^8; y:=1/8*s*(s+4)*(3*s^4-2*s^3-2*s^2+8*s+8)/(s-1)/(s^2+4)/(s+1)^2; t:=1/4*s^5*(s+4)^3/(s-1)/(s^2+4)^2/(s+1)^3; ################################### #icosahedral soln 25: th:=[2/5, 2/5, 1/2, 4/5]; F:=-87808*t^7+263424*t^6-262400*t^5-2560*y*t^4-71552*y^3*t^4+4160*y^3*t^3-1024*y^3*t^2-10976*y^2*t^ 6+27648*y^2*t^5+6880*y^2*t^4+1536*y^2*t^3-5120*y*t^5+21824*y^3*t^5-800*y^4*t^2+55040*y^4*t^3-\ 19404*y^5*t^3-21376*y^5*t^2-1715*y^4*t^5+1235*y^4*t^4+1356*y^5*t^4+18290*y^6*t^2+512*y^7*t^3-\ 4148*y^7*t^2-972*y^7*t-370*y^6*t^3+512*y^8*t^2-243*y^8*t+87808*t^4+243*y^8; y:=1/4*s^2*(5*s^3+2*s^2-4*s-8)*(s+4)^2/(s+1)^2/(s^2+4)/(s-1)/(s^2+3*s+6); t:=1/4*s^5*(s+4)^3/(s-1)/(s^2+4)^2/(s+1)^3; ###################### #icosahedral soln 28 th:=[1/2, 1/2, 1/5, 3/5]; F:= 39062500*t^4-15625000*t^5+3906250*t^6-434375*t^7+54500*t^8-1331*t^9+(-359375000*t^4+109375000*t^5-20831250*t^6+1301500*t^7-95690*t^8)*y+(1507500000*t^4-340875000*t^5+48359625*t^ 6-1736010*t^7+65865*t^8)*y^2+(625000*t^3-3792000000*t^4+615597500*t^5-60888560*t^6+1174500*t^7-13720*t^8)*y^3+(-312500*t^2-3625000*t^3+6329073750*t^4-706544105*t^5+43948580*t^6-\ 406485*t^7)*y^4+(1950000*t^2+7353000*t^3-7314331698*t^4+528011436*t^5-17609682*t^6+65856*t^7)*y^5+(-4726000*t^2-7570160*t^3+5921106495*t^4-250059790*t^5+3383695*t^6)*y^6+( 6213760*t^2+4015040*t^3-3313654080*t^4+68139520*t^5-219520*t^6)*y^7+(-4918080*t^2-42240*t^3+1226301000*t^4-8026200*t^5)*y^8+(20480*t+2154240*t^2-1077120*t^3-270723040*t^4)*y^9+( -4096+8192*t-443136*t^2+439040*t^3+26962544*t^4)*y^10; t := 2*(s^2+5)^3*(s^2-5)^2/(s+5)^3/(s^2-4*s+5)^2/(s+1)^3; y:=(s^5+5*s^4-20*s^3+75*s+75)*(s^2-5)*(s^2+5)/ (s+1)^2/(s^2-4*s+5)/(s+5)/(-75+6*s^2+s^4); ############################################################# # icosahedral soln 29 th:=[1/3, 1/3, 1/3, 4/5]; F:=125*t^7-234*t^6+125*t^5+(-250*t^7+170*t^6+170*t^5-250*t^4)*y+(705*t^6-690*t^5+705*t^4)*y^2+(-\ 960*t^5-960*t^4)*y^3+(15*t^5+3330*t^4+15*t^3)*y^4+(-6*t^5-2010*t^4-2010*t^3-6*t^2)*y^5+(675*t ^4+2010*t^3+675*t^2)*y^6+(-960*t^3-960*t^2)*y^7+720*t^2*y^8+(-80*t^2-80*t)*y^9+(16*t^2-16*t+ 16)*y^10; #{ga = 1/18, be = -1/18, al = 1/50, de = 4/9} y:=1/2*s^2*(s+2)*(s^2+1)*(2*s^2+3*s+3)/(s^2+s+1)/(2+3*s+3*s^2); t:=s^5*(s+2)*(2*s^2+3*s+3)^2/(1+2*s)/(2+3*s+3*s^2)^2; ################################################################3 # icosahedral soln 30 th:=[1/3,1/3,1/3,2/5]; F:=2100875*t^9-545942*t^8+2100875*t^7+(-4201750*t^9-14077290*t^8-14077290*t^7-4201750*t^6)*y+(34674165*t^8+95163030*t^7+34674165*t^6)*y^2+(-3810000*t^8-215538480*t^7-\ 215538480*t^6-3810000*t^5)*y^3+(2985000*t^8+103385310*t^7+554979060*t^6+103385310*t^5+2985000*t^4)*y^4+(-1296000*t^8-46980684*t^7-412355124*t^6-412355124*t^5-\ 46980684*t^4-1296000*t^3)*y^5+(432000*t^8+4306560*t^7+285737370*t^6+186767820*t^5+285737370*t^4+4306560*t^3+432000*t^2)*y^6+(-1724160*t^7-108941280*t^6-108683040*t ^5-108683040*t^4-108941280*t^3-1724160*t^2)*y^7+(56162520*t^6-64271385*t^5+180729090*t^4-64271385*t^3+56162520*t^2)*y^8+(-18279040*t^6+31677970*t^5-31677970*t^4-\ 31677970*t^3+31677970*t^2-18279040*t)*y^9+(3655808*t^6-10967424*t^5+17915169*t^4-17551298*t^3+17915169*t^2-10967424*t+3655808)*y^10; y:=s^4*(s+2)*(2*s^2+3*s+3)*(7*s^2+10*s+7)/(2+3*s+3*s^2)/ (4*s^6+12*s^5+15*s^4+10*s^3+15*s^2+12*s+4); t:=s^5*(s+2)*(2*s^2+3*s+3)^2/(1+2*s)/(2+3*s+3*s^2)^2; ##{al = 9/50, de = 4/9, be = -1/18, ga = 1/18} ##########################################################3 #generic icosahedral soln (soln 33) th:=[2/5, 1/2, 1/3, 4/5]; F:= (15524784*t^2-5373216*t+1350000)*y^(12)- (128381760*t^2-13366080*t)*y^(11)+ (5425704*t^3+496677744*t^2-30539160*t)*y^(10)- (14929920*t^4+41364000*t^3+866759680*t^2-2928160*t)*y^9+ (107546535*t^4-508275750*t^3+747613335*t^2-1837080*t)*y^8- (24385536*t^5-285548724*t^4-2437066824*t^3+74927724*t^2+944784*t)*y^7+ (58212000*t^5-2865570750*t^4-4456260900*t^3+17631810*t^2)*y^6- (49787136*t^6-904003584*t^5-7215732804*t^4-2130570936*t^3-12872196*t^2)*y^5- (413500320*t^6+3724484160*t^5+4839581265*t^4+162430110*t^3+3750705*t^2)*y^4+ (3001304640*t^6+74794560*t^5+2710584000*t^4-380946240*t^3)*y^3- (940800000*t^7+977540640*t^6-726801696*t^5+939255264*t^4-72013536*t^3)*y^2+ (1176000000*t^7-1481095680*t^6+765158400*t^5)*y- (1920800000*t^8-7212800000*t^7+10522980864*t^6-6913299456*t^5+1728324864*t^4); y:=-9*s*(s^2+1)*(3*s-4)*(15*s^4-5*s^3+3*s^2-3*s+2)/ (2*s-1)^2/(9*s^2+4)/(9*s^2+3*s+10); t:=27/4*(s^2+1)^2*(3*s-4)^3*s^5/(9*s^2+4)^2/(2*s-1)^3; ##### # Some elliptic solutions ##### #################################################### # icosahedral soln 27 (2438) th:=[2/5,2/5,2/3,2/5]; F:= 22370117*t^10+23445765*t^9-1479016*t^8-1246824*t^7+1252274*t^6-55566*t^5+(-159279939*t^9-\ 263197620*t^8-2144808*t^7+9283428*t^6-5496318*t^5+111132*t^4)*y+(78656760*t^9+248071320*t^8+ 1455536400*t^7-2231040*t^6-17338800*t^5+8775360*t^4)*y^2+(33614000*t^9-785693636*t^8+ 1163359280*t^7-4805606992*t^6+50828592*t^5+10955528*t^4-7558272*t^3)*y^3+(-289920750*t^8+ 3398719218*t^7-6528219360*t^6+10318895856*t^5-79131756*t^4-3962844*t^3+3779136*t^2)*y^4+( 1019287998*t^7-8296820328*t^6+15326779392*t^5-15248665800*t^4+73713468*t^3-4461480*t^2)*y^5+( -1983777696*t^6+12544666096*t^5-21202892544*t^4+15621486240*t^3-19366096*t^2)*y^6+(-40197060* t^6+2481749280*t^5-12271091760*t^4+18558026640*t^3-10942824600*t^2)*y^7+(-22143375*t^6+ 236727441*t^5-2235811896*t^4+7705356984*t^3-9980508150*t^2+4872106746*t)*y^8+(66430125*t^5-\ 395155908*t^4+1328232168*t^3-2689972092*t^2+2706725970*t-1082690388)*y^9; #{ga = 2/9, be = -2/25, de = 21/50, al = 9/50} y:=1/2+1/30*(350*s^3+63*s^2-6*s-2)/(5*s+4)/(2*s+1)/(8*s+1)*u/s^2; t:=1/2+1/54*(25*s^4+170*s^3+42*s^2+8*s-2)/(5*s+4)^2*u/s^3; #expression for u^2: u2:=s*(8*s+1)*(5*s+4); #################### #icosahedral soln 34: th:=[1/5, 1/3, 1/5, 1/2]; F:= 6912*t^7+1728*t^8+3024*t^9+11664*y^12-6048*t^9*y-42336*t^8*y-91584*t^7*y+ 116424*t^8*y^2+620568*t^7*y^2+32832*t^6*y^2+640*t^5*y^3-93520*t^8*y^3-1931280* t^7*y^3-541920*t^6*y^3+184245*y^4*t^5+7446*t^4*y^5+46200*t^8*y^4+3211965*t^7*y ^4+2331270*t^6*y^4-1088970*t^5*y^5+672*t^8*y^5-3408222*t^7*y^5-4748814*t^6*y^5 -123242*t^4*y^6-224*t^8*y^6+2040374*t^7*y^6+6165978*t^6*y^6+2694650*t^5*y^6-\ 5337054*t^5*y^7+796470*t^4*y^7+38646*t^3*y^7-572208*t^7*y^7-4163742*t^6*y^7+ 821055*t^6*y^8+5809230*t^5*y^8-344520*t^4*y^8-504615*t^3*y^8+405*t^2*y^8-7875* t^7*y^8-3419310*t^5*y^9-575350*t^4*y^9+574240*t^3*y^9+119340*t^2*y^9+1750*t^7* y^9+733250*t^6*y^9+1944*t*y^10-388500*t^6*y^10+794250*t^5*y^10+739350*t^4*y^10 -169050*t^3*y^10-208170*t^2*y^10-326250*t^5*y^11+326250*t^4*y^11-587550*t^3*y^ 11+359100*t^2*y^11-42768*t*y^11+131250*t^6*y^11-48600*t*y^12-21875*t^6*y^12+ 65625*t^5*y^12-82500*t^4*y^12+55625*t^3*y^12+31725*t^2*y^12; y:=1/2+1/2*(5+3*s)*(11-13*s+12*s^2-10*s^3+8*s^4)/(5-15*s+2*s^3)/u; t:=1/2-1/2*(-15+66*s-15*s^2+20*s^3+8*s^6)*u/(5+3*s)/(5-5*s+8*s^2)^2; u2:=(5+3*s)*(5-5*s+8*s^2); ############### #icosahedral soln 35: th:=[2/5,1/3,2/5,1/2]; F:= 1309246092*y^6*t^3-329204736*y^7*t^2+223356187170*y^5*t^5+255052094850*y^4*t^6 -2599210782*y^5*t^4-23058289665*y^4*t^5-162284427520*y^3*t^6+94870381248*y^2*t ^7-186239126016*y*t^8+806215680*y^9*t+94401855*y^8*t^2-18253157382*y^7*t^3+ 88849233774*y^6*t^4+65222466240*y^3*t^7+212675452416*y*t^9+58503674592*y^2*t^8 +9049780824480*t^8*y^4-3008662670720*t^9*y^3+757545255936*t^10*y^2-92703647232 *t^11*y+69711855000*t^4*y^12+382518236250*t^5*y^11-44211328000*t^10+ 46351823616*t^11-12297518496*t*y^10+85109481540*t^2*y^9-312502813425*t^3*y^8-\ 35284073670*t^4*y^7+24755860224*y^12+311625099000*t^6*y^10+28403589250*t^7*y^9 +22999500000*t^8*y^8-4924800000*t^9*y^7-29196100625*t^3*y^12-382518236250*t^4* y^11-856218930750*t^5*y^10-445034354250*t^6*y^9-66136565625*y^12*t^5-\ 219814151625*t^7*y^8-69836400000*t^8*y^7+17236800000*t^9*y^6+168031584*y^5*t^3 -933508800*y^4*t^4+2655313920*y^3*t^5+290605738290*t^5*y^6-3185110466730*t^6*y ^5+3142579854015*t^7*y^4-2377004059200*t^8*y^3+717343586400*t^9*y^2-\ 264567260160*t^10*y+5623876608*y^2*t^6+33764258304*y*t^7+41434883775*t^2*y^12-\ 156302466450*t^3*y^11+1822248615150*t^4*y^10-882174098490*t^5*y^9+ 2661721566345*t^6*y^8-510978744432*t^7*y^7+213666134304*t^8*y^6+10663554048*t^ 8+22045521875*y^12*t^6+77498568000*t^9*y^5-16892064000*t^10*y^4-37859594400*t* y^12+212856408900*t^2*y^11-656518680450*t^3*y^10-1214165537650*t^4*y^9+ 1724652779670*t^5*y^8-5265757029618*t^6*y^7-132273131250*y^11*t^6+16987072512* t^9+2507946042546*t^7*y^6-900445758912*t^8*y^5-152378100000*t^9*y^4+ 33784128000*t^10*y^3-221351133888*t*y^11+1025048190330*t^2*y^10-3019234545360* t^3*y^9+8376999528060*t^4*y^8-13701277887570*t^5*y^7+19754801651970*t^6*y^6-\ 15819508647738*t^7*y^5-5035261952*t^7; t:=1/2+1/2*(8*s^6+20*s^3-15*s^2+66*s-15)/u/(8*s^2-5*s+5); y:=1/2+ 1/2/u/(26*s^3+60*s^2+15*s+35)*(3*s+5)*(16*s^5-8*s^4+18*s^3-8*s^2+115*s+3); u2:=(3*s+5)*(8*s^2-5*s+5); ########################### # icosahedral soln 36: th:=[1/3,1/5,1/3,2/5]; F:= -1650000*t^7*y^2-250000*t^9+67600000*y^12-576000*t^5*y^4+1688000*t^6*y^3+82944 *t^4*y^5+33566452875*t^8*y^4+75222375*t^6*y^4-8400000*t^10*y^5+33426750*t^9*y^ 5-714951552*t^8*y^5-53361051288*t^7*y^5+552133176*t^6*y^5-40440030*t^5*y^5+ 2800000*t^10*y^6+25947250*t^9*y^6+184010260*t^8*y^6+1358810432*t^7*y^6+ 61506459644*t^6*y^6-627927874*t^5*y^6-265514250*t^7*y^4+2204496*t^6*y^12-\ 6613488*t^5*y^12-50000*t^8+12300288*t^4*y^6-1557504*t^3*y^7+8891613*t^4*y^12-\ 6760746*t^3*y^12+13078125*t^2*y^12-10800000*t*y^12-11413500*t^9*y^7-56907270*t ^8*y^7-562917840*t^7*y^7-1272647856*t^6*y^7-52067611692*t^5*y^7+433855662*t^4* y^7+41763645*t^8*y^8+31333050*t^7*y^8+754008585*t^6*y^8+592417620*t^5*y^8+ 32218988580*t^4*y^8-9797760*t^8*y^9-45310860*t^7*y^9+59085180*t^6*y^9+5009375* t^10-176511480*t^3*y^8+38000000*t^2*y^9-687694030*t^5*y^9+44364760*t^4*y^9-\ 14270647290*t^3*y^9+16411248*t^7*y^10-13289346*t^6*y^10-11369970*t^5*y^10+ 343640226*t^4*y^10-179026710*t^3*y^10+4306875000*t^2*y^10+2204496*t^6*y^11+ 7690464*t^5*y^11-11952198*t^4*y^11-79540020*t^3*y^11+68006250*t^2*y^11-3600000 *t*y^10+1959552*t^8*y^10-793200000*t*y^11-4408992*t^7*y^11+1350000*t^8*y-\ 803906250*t^11*y+19162500*t^10*y-27806250*t^9*y-16406250*t^11*y^2+4442943750*t ^10*y^2-30318750*t^9*y^2+67031250*t^8*y^2+10937500*t^11*y^3+36268750*t^10*y^3-\ 14905338000*t^9*y^3+77807750*t^8*y^3-93364000*t^7*y^3-41212500*t^10*y^4+ 127627500*t^9*y^4+68359375*t^12-5468750*t^11; y:=1/2+1/18/u/(s+1)/(7*s^3-3*s^2-s+1)* (140*s^6+1029*s^5-1023*s^4+360*s^3-288*s^2+27*s+27); t:=1/2+1/6*(40*s^6+540*s^5-765*s^4+540*s^3-270*s^2+27)/u/(8*s^2-9*s+3)/(s+1)^2; u2:=3*(5*s+1)*(8*s^2-9*s+3); ########################### # icosahedral soln 37: # (15 branch Valentiner solution) th:=[1/3,1/3,1/3,1/5]; F1:= 328261718750*t^17-8873010156250*t^16-27000291253125*t^15-60280806115625*t^14-\ 60280806115625*t^13-27000291253125*t^12-8873010156250*t^11+328261718750*t^10+( 23664675000000*t^16+211300667831250*t^15+733943391300000*t^14+936957905925000* t^13+733943391300000*t^12+211300667831250*t^11+23664675000000*t^10)*y+(-\ 37793210156250*t^16-594901895043750*t^15-3153060441478125*t^14-\ 6275958262978125*t^13-6275958262978125*t^12-3153060441478125*t^11-\ 594901895043750*t^10-37793210156250*t^9)*y^2+(36181801562500*t^16+ 888776581400000*t^15+8467349083417000*t^14+19660503544044500*t^13+ 29095897662839500*t^12+19660503544044500*t^11+8467349083417000*t^10+ 888776581400000*t^9+36181801562500*t^8)*y^3+(-16479492187500*t^16-\ 942319365168750*t^15-13685474302147500*t^14-47863845677154375*t^13-\ 68294160688873125*t^12-68294160688873125*t^11-47863845677154375*t^10-\ 13685474302147500*t^9-942319365168750*t^8-16479492187500*t^7)*y^4+( 6591796875000*t^16+592621009050000*t^15+16896186697803000*t^14+ 74058995075056986*t^13+148651220648470056*t^12+95118799457827416*t^11+ 148651220648470056*t^10+74058995075056986*t^9+16896186697803000*t^8+ 592621009050000*t^7+6591796875000*t^6)*y^5+(-320631156662500*t^15-\ 14838047723562750*t^14-90897714423306180*t^13-213628384699376835*t^12-\ 159923580257372985*t^11-159923580257372985*t^10-213628384699376835*t^9-\ 90897714423306180*t^8-14838047723562750*t^7-320631156662500*t^6)*y^6+( 90439453125000*t^15+10837582525696500*t^14+80506305809892360*t^13+ 262146492001128960*t^12+131414942424621420*t^11+263287111097509020*t^10+ 131414942424621420*t^9+262146492001128960*t^8+80506305809892360*t^7+ 10837582525696500*t^6+90439453125000*t^5)*y^7+(-22609863281250*t^15-\ 5796137285156250*t^14-59380262518503735*t^13-236887208640116235*t^12-\ 108788129298050760*t^11-205764970158110520*t^10-205764970158110520*t^9-\ 108788129298050760*t^8-236887208640116235*t^7-59380262518503735*t^6-\ 5796137285156250*t^5-22609863281250*t^4)*y^8; F:=F1+(2463140468750000*t^14+ 32220996397634930*t^13+188782138303147080*t^12+5686114912304730*t^11+ 248950398227673480*t^10+3011139901542060*t^9+248950398227673480*t^8+ 5686114912304730*t^7+188782138303147080*t^6+32220996397634930*t^5+ 2463140468750000*t^4)*y^9+(-682456113281250*t^14-14004464393589486*t^13-\ 115462386925670331*t^12+26478364409053209*t^11-145284809797189656*t^10-\ 38809262135491236*t^9-38809262135491236*t^8-145284809797189656*t^7+ 26478364409053209*t^6-115462386925670331*t^5-14004464393589486*t^4-\ 682456113281250*t^3)*y^10+(124082929687500*t^14+3979998356250000*t^13+ 61106724782719680*t^12-54409408904100420*t^11+145025668671558480*t^10-\ 121603752038657520*t^9+193157931456147060*t^8-121603752038657520*t^7+ 145025668671558480*t^6-54409408904100420*t^5+61106724782719680*t^4+ 3979998356250000*t^3+124082929687500*t^2)*y^11+(-808096477343750*t^13-\ 23823859977531250*t^12+33524593229708515*t^11-70902503235239235*t^10+ 55461555110232240*t^9-37052448491670270*t^8-37052448491670270*t^7+ 55461555110232240*t^6-70902503235239235*t^5+33524593229708515*t^4-\ 23823859977531250*t^3-808096477343750*t^2)*y^12+(7827210307875000*t^12-\ 19611569030156250*t^11+48497092292979360*t^10-76175799691484790*t^9+ 103568196490999920*t^8-108086833121113980*t^7+103568196490999920*t^6-\ 76175799691484790*t^5+48497092292979360*t^4-19611569030156250*t^3+ 7827210307875000*t^2)*y^13+(-1437387687093750*t^12+4716941434406250*t^11-\ 11792353586015625*t^10+17646431591309445*t^9-17562234015881460*t^8+ 6991214576181390*t^7+6991214576181390*t^6-17562234015881460*t^5+ 17646431591309445*t^4-11792353586015625*t^3+4716941434406250*t^2-\ 1437387687093750*t)*y^14+(191651691612500*t^12-1149910149675000*t^11+ 4437729249450000*t^10-11647803208562500*t^9+22678270901003736*t^8-\ 33475275999064944*t^7+38122326722084916*t^6-33475275999064944*t^5+ 22678270901003736*t^4-11647803208562500*t^3+4437729249450000*t^2-\ 1149910149675000*t+191651691612500)*y^15; y:= 1/2-1/4*(4+58*s+2425*s^7+244*s^2+1000*s^8+3805*s^5+874*s^3+1999*s^4+4171*s^6)/ (s+2)/(25*s^6+135*s^5+111*s^4+91*s^3+36*s^2+6*s+1)/u; t:= 1/2-3/2*(500*s^7+925*s^6+1164*s^5+830*s^4+340*s^3+105*s^2+20*s+4)/(s+2)^2/(5*s +1)/u^3; u2:=(1+s+4*s^2)*(5*s+1); ########################### # icosahedral soln 38: # (15 branch Valentiner solution) th:=[1/3,1/3,1/3,3/5]; F:= -9765625*t^13+19921875*t^12-17659375*t^11-17659375*t^10+19921875*t^9-9765625*t ^8+(86718750*t^12-133106250*t^11+317868750*t^10-133106250*t^9+86718750*t^8)*y+ (1171875*t^12-411215625*t^11-377784375*t^10-377784375*t^9-411215625*t^8+ 1171875*t^7)*y^2+(-781250*t^12+38743750*t^11+2817204250*t^10+1117510250*t^9+ 2817204250*t^8+38743750*t^7-781250*t^6)*y^3+(-17278125*t^11-1808780625*t^10-\ 8415706875*t^9-8415706875*t^8-1808780625*t^7-17278125*t^6)*y^4+(300000*t^11+ 755550750*t^10+7864758246*t^9+27822550758*t^8+7864758246*t^7+755550750*t^6+ 300000*t^5)*y^5+(-100000*t^11+4873375*t^10-3637362605*t^9-33920551395*t^8-\ 33920551395*t^7-3637362605*t^6+4873375*t^5-100000*t^4)*y^6+(-1235250*t^10-\ 222019290*t^9+22989536730*t^8+51032654370*t^7+22989536730*t^6-222019290*t^5-\ 1235250*t^4)*y^7+(55320165*t^9-6026433255*t^8-42311496285*t^7-42311496285*t^6-\ 6026433255*t^5+55320165*t^4)*y^8+(1152480*t^9+1345276090*t^8+13953333730*t^7+ 44506756650*t^6+13953333730*t^5+1345276090*t^4+1152480*t^3)*y^9+(-230496*t^9-\ 38907783*t^8-2605518171*t^7-19887227925*t^6-19887227925*t^5-2605518171*t^4-\ 38907783*t^3-230496*t^2)*y^10+(7262730*t^8-632180910*t^7+6461893230*t^6+ 8809581150*t^5+6461893230*t^4-632180910*t^3+7262730*t^2)*y^11+(100521665*t^7-\ 181329755*t^6-3333113785*t^5-3333113785*t^4-181329755*t^3+100521665*t^2)*y^12+ (236379210*t^6-1080750270*t^5+3264398370*t^4-1080750270*t^3+236379210*t^2)*y^ 13+(-112546875*t^6+390728145*t^5-390728145*t^4-390728145*t^3+390728145*t^2-\ 112546875*t)*y^14+(15006250*t^6-45018750*t^5+34401246*t^4+6228758*t^3+34401246 *t^2-45018750*t+15006250)*y^15; y:= 1/2-1/2*(250*s^6+500*s^5+518*s^4+261*s^3+76*s^2+13*s+2)/(s+2)/(5*s+1)/(1+3*s+6 *s^2+5*s^3)/u; t:= 1/2-3/2*(500*s^7+925*s^6+1164*s^5+830*s^4+340*s^3+105*s^2+20*s+4)/ (s+2)^2/(5*s+1)/u^3; u2:=(1+s+4*s^2)*(5*s+1); ########################### # icosahedral soln 39: th:=[1/3, 4/5, 1/3, 4/5]; F:= -71124480*t^5*y-387694125*t^7*y-105043750*t^10+545437500*t^9-1132346875*t^8-\ 610265600*t^6+126794752*t^5+1175424000*t^7-278182500*t^8*y^2-267100350*t^6*y^2 -67359150*t^6*y^6-94770*t^3*y^14+58320*t^4*y^14+349650*t^4*y^12-34121250*t^7*y ^9+27371250*t^6*y^9+27*y^15-12748610*t^5*y^6+12688760*t^4*y^6+532635*t^3*y^6+ 111240*t^2*y^13+79480200*t^6*y^7-15525*t*y^13+159900750*t^6*y^5+10023750*t^7*y ^10+270643200*t^6*y+6788250*t^5*y^7-59656500*t^7*y^7-8290710*t^4*y^7-64146675* t^6*y^8+33792500*t^8*y^6+33229000*t^7*y^6-21623754*t^5*y^5-150639000*t^7*y^5-\ 14035656*t^4*y^5+119079*t^3*y^5-1512*t*y^15-2916*t^3*y^15-408900*t^3*y^7+ 138915*t^2*y^7+4374*t^2*y^15+1458000*t^6*y^12-18225000*t^9*y^5+44422500*t^8*y^ 5-18225000*t^8*y^7-437400*t^5*y^13+38085120*t^5*y^2+421626375*t^7*y^2+15805440 *t^4*y^2+405642275*t^6*y^3-101661930*t^5*y^3-535595000*t^7*y^3-7738880*t^4*y^3 -263067750*t^6*y^4-1822500*t^7*y^11-4483080*t^5*y^8+4221900*t^4*y^8-242070*t^3 *y^8+7841015*t^5*y^9-357500*t^4*y^9-397710*t^3*y^12+156185*t^2*y^12-22940*t*y^ 12-60656250*t^9*y+248831250*t^8*y+124920*t^2*y^8+782595*t^4*y^11+69768750*t^9* y^2+685260*t^4*y^13-520710*t^3*y^11+132720*t^2*y^11+3777300*t^5*y^11+49596690* t^5*y^4+15466665*t^4*y^4-17010*t*y^11-11053125*t^5*y^10-1530900*t^5*y^12+ 4556250*t^8*y^8+60142500*t^7*y^8-2369250*t^6*y^11-76887500*t^9*y^3+316228750*t ^8*y^3-1019140*t^3*y^9+150490*t^2*y^9-634500*t^6*y^10+1850166*t^4*y^10-169608* t^3*y^10+45562500*t^9*y^4-169455000*t^8*y^4+321933750*t^7*y^4-346410*t^3*y^13-\ 8748*t*y^10+73146*t^2*y^10+44280*t^2*y^14-7425*t*y^14; y:=1/2+1/6*(14*s^5+61*s^4-66*s^3-660*s^2-900*s-225)/(s+1)/(s^2-5)/u; t:= 1/2-1/18*(2*s^7+10*s^6-90*s^4-135*s^3+297*s^2+945*s+675)/(4*s^2+15*s+15)^2/ (s^2-5)*u; u2:=3*(s+5)*(4*s^2+15*s+15); ########################### # icosahedral soln 40: th:= [3/5, 2/3, 3/5, 2/3]; F:= 458064650136528*t^15-703331388283131*t^14+721058367621096*t^13-493635300323841 *t^12+217639434252186*t^11-56082971126871*t^10+6436093853696*t^9+528125*y^15+ 108229420522260*y*t^14-147724827816375*y*t^13+129297219097770*y*t^12-\ 70950231322965*y*t^11+22308167517660*y*t^10-3069789260655*y*t^9+56143841163120 *y^2*t^14-82054469043945*y^2*t^13+83613983813910*y^2*t^12-60705216410175*y^2*t ^11+30813067658700*y^2*t^10-9854296118205*y^2*t^9+1457655475770*y^2*t^8-\ 41197500788040*y^3*t^14+58238067026610*y^3*t^13-57781015309215*y^3*t^12+ 44899891553610*y^3*t^11-28381615818885*y^3*t^10+12553699793480*y^3*t^9-\ 2914820503955*y^3*t^8+221947816610*y^3*t^7-16132952665530*y^4*t^14+ 42515902770120*y^4*t^13-63122876194230*y^4*t^12+52176050537670*y^4*t^11-\ 21504664620945*y^4*t^10+2818112442510*y^4*t^9+457442741265*y^4*t^8-\ 209957014680*y^4*t^7+44193149325*y^4*t^6+16152545714220*y^5*t^14-\ 40340041559154*y^5*t^13+55692781178616*y^5*t^12-39241879068258*y^5*t^11+ 8131390911882*y^5*t^10+5212228890075*y^5*t^9-3265749321432*y^5*t^8+ 942202736637*y^5*t^7-189810002970*y^5*t^6+15909533757*y^5*t^5+19587345147090*y ^6*t^13-27751959273960*y^6*t^12+18492034403430*y^6*t^11-2515773330540*y^6*t^10 -1348979487435*y^6*t^9-1104126467140*y^6*t^8+859371394315*y^6*t^7-208873684270 *y^6*t^6+23218807755*y^6*t^5-4115902288500*y^7*t^13+4951653731550*y^7*t^12+ 816075905580*y^7*t^11-6737356024470*y^7*t^10+4157299276470*y^7*t^9+24463111275 *y^7*t^8-372678312960*y^7*t^7+123598117125*y^7*t^6-28244112210*y^7*t^5+ 2823683265*y^7*t^4+938752035300*y^8*t^12-4014861182640*y^8*t^11+5832061707825* y^8*t^10-2979696033180*y^8*t^9+217019887245*y^8*t^8-21644435970*y^8*t^7+ 3012577155*y^8*t^6+4537733400*y^8*t^5-451524510*y^8*t^4-598304151000*y^9*t^12+ 2021974360020*y^9*t^11-3100826699100*y^9*t^10+2118455826165*y^9*t^9-\ 896337125010*y^9*t^8+476777924985*y^9*t^7-105084738340*y^9*t^6+2359954275*y^9* t^5+2464738380*y^9*t^4-380218750*y^9*t^3+36088780500*y^10*t^12-180796009500*y^ 10*t^11+210657559248*y^10*t^10+196898917635*y^10*t^9-335865226914*y^10*t^8+ 123332104971*y^10*t^7-51154153716*y^10*t^6+7148709729*y^10*t^5-1757350170*y^10 *t^4+1330296342*y^10*t^3-238050000*y^10*t^2-6743614500*y^11*t^11+80904784500*y ^11*t^10-282369867750*y^11*t^9+334405108125*y^11*t^8-193017548250*y^11*t^7+ 65248865625*y^11*t^6+7645581000*y^11*t^5-12046478625*y^11*t^4+3888216750*y^11* t^3-481218750*y^11*t^2+2706412500*y^12*t^11-15050387250*y^12*t^10+57077055000* y^12*t^9-78444342000*y^12*t^8+66735006750*y^12*t^7-49129544875*y^12*t^6+ 25967718250*y^12*t^5-10801688125*y^12*t^4+2257826000*y^12*t^3-51359375*y^12*t^ 2-42250000*y^12*t-1940962500*y^13*t^10+4332993750*y^13*t^9-15928650000*y^13*t^ 8+29025112500*y^13*t^7-27048131250*y^13*t^6+13568690625*y^13*t^5-2076450000*y^ 13*t^4-907490625*y^13*t^3+502968750*y^13*t^2-75609375*y^13*t+478406250*y^14*t^ 9+236925000*y^14*t^8-2101950000*y^14*t^7+3515325000*y^14*t^6-3828440625*y^14*t ^5+3069937500*y^14*t^4-1722759375*y^14*t^3+538218750*y^14*t^2-68390625*y^14*t+ 9112500*y^15*t^9-41006250*y^15*t^8-119475000*y^15*t^7+609525000*y^15*t^6-\ 1018993750*y^15*t^5+927990625*y^15*t^4-508000000*y^15*t^3+166109375*y^15*t^2-\ 26318750*y^15*t-109350000*t^10*y^14+492075000*t^11*y^13-984150000*t^12*y^12+ 738112500*t^13*y^11-1459469846250*t^17*y+12908815264080*t^16*y-49539312077400* t^15*y-5130328403538*t^18+47048637539592*t^17-192067194737592*t^16+ 6967343856420*t^16*y^2-875681907750*t^17*y^2-25506173034720*t^15*y^2+ 18687965694660*t^15*y^3+583787938500*t^17*y^3-4910647700250*t^16*y^3-\ 291495388500*t^16*y^4+3250965133620*t^15*y^4+318864600000*t^16*y^5-\ 3430029572748*t^15*y^5+1361817562500*t^15*y^6-106288200000*t^16*y^6-\ 7285143606120*t^14*y^6+1107168750*y^11*t^12-4059618750*y^10*t^13+76256862750*y ^9*t^13-56229410250*y^8*t^14+80897130000*y^8*t^13-146146275000*y^7*t^15+ 1321014703500*y^7*t^14; y:= 1/2-1/2* (2*s^9+20*s^8+53*s^7-89*s^6-605*s^5-851*s^4-1389*s^3-5775*s^2-10125*s-5625)/ (s^2-5)/(s^2-6*s-15)/(s^2+4*s+5)/u; t:= 1/2-1/18*(2*s^7+10*s^6-90*s^4-135*s^3+297*s^2+945*s+675)/(4*s^2+15*s+15)^2/ (s^2-5)*u; u2:=3*(s+5)*(4*s^2+15*s+15); ########################### # icosahedral soln 41: # Equivalent to Dubrovin--Mazzocco's (10 page) elliptic icosahedral soln th:=[1/3,1/3,1/3,1/3]; F:= 16807*t^14+40716*t^13+62101*t^12+40716*t^11+16807*t^10+(-323832*t^13-739050*t^ 12-739050*t^11-323832*t^10)*y+(16932*t^13+2875320*t^12+4844316*t^11+2875320*t^ 10+16932*t^9)*y^2+(3292*t^13-132840*t^12-12625036*t^11-12625036*t^10-132840*t^ 9+3292*t^8)*y^3+(-21870*t^13-254676*t^12-3846972*t^11+18875856*t^10-3846972*t^ 9-254676*t^8-21870*t^7)*y^4+(8748*t^13+553560*t^12+6523764*t^11+52435320*t^10+ 52435320*t^9+6523764*t^8+553560*t^7+8748*t^6)*y^5+(-315740*t^12-7897860*t^11-\ 58020372*t^10-254421104*t^9-58020372*t^8-7897860*t^7-315740*t^6)*y^6+(96228*t^ 12+5381376*t^11+52933272*t^10+217938444*t^9+217938444*t^8+52933272*t^7+5381376 *t^6+96228*t^5)*y^7+(-24057*t^12-2515050*t^11-34813191*t^10-141713832*t^9+ 206140134*t^8-141713832*t^7-34813191*t^6-2515050*t^5-24057*t^4)*y^8+(911736*t^ 11+17141202*t^10+62633474*t^9-587326832*t^8-587326832*t^7+62633474*t^6+ 17141202*t^5+911736*t^4)*y^9+(-192456*t^11-6852888*t^10-14646444*t^9+725440188 *t^8+1024377216*t^7+725440188*t^6-14646444*t^5-6852888*t^4-192456*t^3)*y^10+( 34992*t^11+1705860*t^10+2031540*t^9-616521768*t^8-990076680*t^7-990076680*t^6-\ 616521768*t^5+2031540*t^4+1705860*t^3+34992*t^2)*y^11+(-316386*t^10+747954*t^9 +374703982*t^8+708386816*t^7+734623128*t^6+708386816*t^5+374703982*t^4+747954* t^3-316386*t^2)*y^12+(128304*t^9-172879596*t^8-366892008*t^7-412698972*t^6-\ 412698972*t^5-366892008*t^4-172879596*t^3+128304*t^2)*y^13+(56501064*t^8+ 153720504*t^7+142005552*t^6+209624280*t^5+142005552*t^4+153720504*t^3+56501064 *t^2)*y^14+(-12754584*t^8-47795616*t^7-36521496*t^6-60234840*t^5-60234840*t^4-\ 36521496*t^3-47795616*t^2-12754584*t)*y^15+(1594323*t^8+11691702*t^7+2623806*t ^6+18930984*t^5+4188669*t^4+18930984*t^3+2623806*t^2+11691702*t+1594323)*y^16+ (-2125764*t^7+1062882*t^6-3715926*t^5-535602*t^4-535602*t^3-3715926*t^2+ 1062882*t-2125764)*y^17+(708588*t^6-2125764*t^5+4339408*t^4-5135876*t^3+ 4339408*t^2-2125764*t+708588)*y^18; u2:=(8*s^2-11*s+8)*s; # so u^2=u2 is our elliptic curve y:=(1/2) - (8*s^7-28*s^6+75*s^5+31*s^4-269*s^3+318*s^2-166*s+56)/ (18*u*(s-1)*(3*s^3-4*s^2+4*s+2)); t:=(1/2) + (s+1)*(32*s^8-320*s^7+1112*s^6-2420*s^5+ 3167*s^4-2420*s^3+1112*s^2-320*s+32)/ (54*u^3*s*(s-1)); ################################################ # icosahedral soln 42: # genus one, 20 branches th:=[1/3, 1/2, 1/3, 4/5]; F:= 13468840704*t^4*y^20-134688407040*t^4*y^19-26937681408*t^3*y^20+2102132736*t^7 *y^15-21741112800*t^6*y^16+429489941760*t^5*y^17+323977864320*t^4*y^18+ 322012247040*t^3*y^19+8205297408*t^2*y^20-15765995520*t^7*y^14+166571437824*t^ 6*y^15-3585441166560*t^5*y^16-150906032640*t^4*y^17-1147992341760*t^3*y^18-\ 161006123520*t^2*y^19+5263543296*t*y^20-3649536000*t^9*y^11+92200096395*t^8*y^ 12-692776093680*t^7*y^13+21246312261240*t^6*y^14-19213596705600*t^5*y^15+ 22495166171280*t^4*y^16-6278221889280*t^3*y^17+2381850852480*t^2*y^18-\ 212941716480*t*y^19+9331200000*y^20+20072448000*t^9*y^10-536777666370*t^8*y^11 +4373361822060*t^7*y^12-148956958402560*t^6*y^13+149628471256440*t^5*y^14-\ 98519089429728*t^4*y^15+23582184341760*t^3*y^16-4625052410880*t^2*y^17+ 215091624960*t*y^18+4400000000*t^11*y^7-113426500000*t^10*y^8+4150470513750*t^ 9*y^9-38933666842246*t^8*y^10+58765700271080*t^7*y^11+424104661547490*t^6*y^12 -371340519362560*t^5*y^13+174659045165480*t^4*y^14-26640676096128*t^3*y^15+ 2739025473840*t^2*y^16-12877608960*t*y^17-15400000000*t^11*y^6+429506000000*t^ 10*y^7-18260528171875*t^9*y^8+180610592121080*t^8*y^9-208357418463040*t^7*y^10 -866349622198060*t^6*y^11+589804879503620*t^5*y^12-194567908678720*t^4*y^13+ 16123132555880*t^3*y^14-466478003808*t^2*y^15+470292480*t*y^16+137500000000*t^ 12*y^4-5763725000000*t^11*y^5+49727555900000*t^10*y^6-65043622666000*t^9*y^7-\ 267935723114785*t^8*y^8+152749492135460*t^7*y^9+1458286845384676*t^6*y^10-\ 694719391147640*t^5*y^11+156186065390315*t^4*y^12-7801344790480*t^3*y^13+ 36116756480*t^2*y^14+241864704*t*y^15-275000000000*t^12*y^3+13622812500000*t^ 11*y^4-118816051200000*t^10*y^5+59501468435750*t^9*y^6+491498738590090*t^8*y^7 -59959686483950*t^7*y^8-1663755462971960*t^6*y^9+500772188447600*t^5*y^10-\ 64473426530530*t^4*y^11+890445560120*t^3*y^12+4883328000*t^2*y^13+ 6875000000000*t^13*y-9625000000000*t^12*y^2-4707450000000*t^11*y^3+ 112316443800000*t^10*y^4+152889051851250*t^9*y^5-647640763946960*t^8*y^6-\ 178415905875080*t^7*y^7+1444902639647950*t^6*y^8-240632395956490*t^5*y^9+ 12217770688274*t^4*y^10+48814807520*t^3*y^11-349920000*t^2*y^12-3437500000000* t^13-34925000000000*t^12*y+52830662500000*t^11*y^2-97202628600000*t^10*y^3-\ 265421398941875*t^9*y^4+359912636282296*t^8*y^5+481809839279980*t^7*y^6-\ 962976272903780*t^6*y^7+76973367698785*t^5*y^8-391857841840*t^4*y^9-\ 10604463264*t^3*y^10+19360000000000*t^12+69911600000000*t^11*y-69698741200000* t^10*y^2+267774143992000*t^9*y^3-9076308852500*t^8*y^4-434565752983796*t^7*y^5 +419320241128480*t^6*y^6-8874572271480*t^5*y^7-155379076125*t^4*y^8+810000000* t^3*y^9-45038950000000*t^11-77436108200000*t^10*y+1842890875000*t^9*y^2-\ 181010693492000*t^8*y^3+196084660528750*t^7*y^4-98626728581000*t^6*y^5-\ 1025628797250*t^5*y^6+23105126250*t^4*y^7+56173046200000*t^10+56930856700000*t ^9*y+33713152075000*t^8*y^2+3313447600000*t^7*y^3-2442227612500*t^6*y^4+ 299644831250*t^5*y^5-40177733750000*t^9-26795160000000*t^8*y-7027426875000*t^7 *y^2+1470612500000*t^6*y^3-11817421875*t^5*y^4+15756562500000*t^8+ 5252187500000*t^7*y-262609375000*t^6*y^2-2626093750000*t^7; t:= 1/2-1/2/u^3*(s+3)/(s^2+1)^2* (8*s^10+100*s^7-135*s^6+834*s^5-1205*s^4+2280*s^3-1365*s^2+890*s+321); y:=1/2+ 1/6/u/(s^2+1)/(3*s^2-4*s+5)* (41-196*s+214*s^2+85*s^4-196*s^3-28*s^5+8*s^6)*(s+3); u2:=3*(8*s^2-13*s+17)*(s+3); ################################################ # icosahedral soln 43: # genus one, 20 branches th:=[1/3, 1/2, 1/3, 2/5]; F:= 216721397537533485851904*t^12*y^20-2167213975375334858519040*t^12*y^19-\ 1300328385225200915111424*t^11*y^20-4645091682474568884000*t^14*y^16+ 38038924302414956593920*t^13*y^17+11242261895910345769622400*t^12*y^18+ 11326818886838068723799040*t^11*y^19+3753760452687184415973120*t^10*y^20+ 37160733459796551072000*t^14*y^15-290815214793205148860320*t^13*y^16-\ 39662011772961765676669440*t^12*y^17-51527154204029640558240000*t^11*y^18-\ 28317047217095171809497600*t^10*y^19-6849125398871580358010880*t^9*y^20+ 69139191375805078125*t^16*y^12-926813850025430250000*t^15*y^13-\ 152026553727912031515000*t^14*y^14+1223889574413206135078208*t^13*y^15+ 104961069628279229381765040*t^12*y^16+160107552522885826032134400*t^11*y^17+ 111898302589204934600615040*t^10*y^18+44595850806927722267550720*t^9*y^19+ 8721154526421897465391872*t^8*y^20-414835148254830468750*t^16*y^11+ 5471176494158856000000*t^15*y^12+420824144424135303720000*t^14*y^13-\ 3599307307232438102795160*t^13*y^14-219124295035922052512382240*t^12*y^15-\ 374464355007787102339142400*t^11*y^16-302805360816281453437747200*t^10*y^17-\ 152137250658360819840637440*t^9*y^18-48836410769497651374159360*t^8*y^19-\ 8093477949935317779727872*t^7*y^20+7867454639941406250*t^17*y^9+ 1280765124726517968750*t^16*y^10-18552206177154407754000*t^15*y^11-\ 852196459707898918039980*t^14*y^12+7949097411015346226382240*t^13*y^13+ 371879320938172773745265640*t^12*y^14+691928113383008768911289472*t^11*y^15+ 618403757656068636933606480*t^10*y^16+354779759684830638515155200*t^9*y^17+ 143373386951344852874501760*t^8*y^18+38661638863623160230743040*t^7*y^19+ 5576227055228888053274880*t^6*y^20-35403545879736328125*t^17*y^8-\ 2668043462402812500000*t^16*y^9+43776569222931376647000*t^15*y^10+ 1316581207623366621185880*t^14*y^11-13727335722956675698889520*t^13*y^12-\ 522587565559928350396103040*t^12*y^13-1038616485801799815339697080*t^11*y^14-\ 999804548296471351801453920*t^10*y^15-623162227760720949082344480*t^9*y^16-\ 286296959164588876865733120*t^8*y^17-97281622383351796182216960*t^7*y^18-\ 22346338134163287903782400*t^6*y^19-2864222778696479368197120*t^5*y^20+ 1432399570312500000*t^18*y^6+93039779735943750000*t^17*y^7+ 3788145760367629828125*t^16*y^8-74516295540249615195000*t^15*y^9-\ 1596983206275528542920296*t^14*y^10+19005317055709902691261200*t^13*y^11+ 614954849512381749205083270*t^12*y^12+1287598695290329011665307840*t^11*y^13+ 1314264300022387360657485720*t^10*y^14+864990338484034968408666432*t^9*y^15+ 427486844559735625564639440*t^8*y^16+165617233316315923221538560*t^7*y^17+ 47734341349349802337752960*t^6*y^18+9350111186048849078983680*t^5*y^19+ 1088880897731635924199424*t^4*y^20-4297198710937500000*t^18*y^5-\ 173314277769846093750*t^17*y^6-3695866652338277906250*t^16*y^7+ 95298337254334471087500*t^15*y^8+1545960869947639827640080*t^14*y^9-\ 21409364995510662869526720*t^13*y^10-609758094479127189312345300*t^12*y^11-\ 1330986354711495364908338400*t^11*y^12-1426159941795196377477174720*t^10*y^13-\ 974207521028128410520040520*t^9*y^14-500499528610492189071431136*t^8*y^15-\ 208543369373476955933820480*t^7*y^16-68769083239611061020595200*t^6*y^17-\ 16628140364550019392015360*t^5*y^18-2759154581540753405337600*t^4*y^19-\ 300003426889121294843904*t^3*y^20+10643915823046875000*t^18*y^4+ 188189554601861718750*t^17*y^5+2592666050462163900000*t^16*y^6-\ 94420700153868831974000*t^15*y^7-1206054418126749180677660*t^14*y^8+ 19782408711571334077341180*t^13*y^9+510639191635452376583699748*t^12*y^10+ 1152255772926514778573252880*t^11*y^11+1287973943444562617763671220*t^10*y^12+ 903033830093845940508664800*t^9*y^13+471616335253942786499378040*t^8*y^14+ 203083920518736026750955264*t^7*y^15+72383397265321406239248720*t^6*y^16+ 19939899073139891108858880*t^5*y^17+3945349945439539329968640*t^4*y^18+ 545061972085630154219520*t^3*y^19+57297309741598579200000*t^2*y^20-\ 14125833794531250000*t^18*y^3-151133727542783203125*t^17*y^4-\ 1152223226974363950000*t^16*y^5+72982366663608203788000*t^15*y^6+ 763801699685849918048840*t^14*y^7-15028710394581080353444630*t^13*y^8-\ 360818411309331630101581200*t^12*y^9-835533689120752387890496752*t^11*y^10-\ 970982276104852405204399560*t^10*y^11-693564594708826076407709040*t^9*y^12-\ 362467853939950810018894080*t^8*y^13-156665668129284921431086440*t^7*y^14-\ 57870563973741609531922080*t^6*y^15-17243407774510497008006400*t^5*y^16-\ 3811322495953903601111040*t^4*y^17-589832089807747163351040*t^3*y^18-\ 64789870736294784000000*t^2*y^19-6883699731038208000000*t*y^20+ 135900351562500000*t^19*y+10842001400390625000*t^18*y^2+91306266571575000000*t ^17*y^3+273156623101237300000*t^16*y^4-44614766785085297495000*t^15*y^5-\ 393486677856127810357960*t^14*y^6+9355375953923319763260240*t^13*y^7+ 214226859416866010572595190*t^12*y^8+505491906404086248314632440*t^11*y^9+ 610237047569957280159587000*t^10*y^10+441901268426776291036246640*t^9*y^11+ 228360305646014717893820565*t^8*y^12+96790853177369820855902160*t^7*y^13+ 35811621427364469644112000*t^6*y^14+11084431320935954656657920*t^5*y^15+ 2612546526909349000753920*t^4*y^16+430935461242177839575040*t^3*y^17+ 47256831785756160000000*t^2*y^18+3514934606856192000000*t*y^19+ 397894913910374400000*y^20-67950175781250000*t^19-5786337628125000000*t^18*y-\ 45109147998037500000*t^17*y^2+32757146109189400000*t^16*y^3+ 21159377547128964357500*t^15*y^4+166570861080021236642096*t^14*y^5-\ 4731914305225256071091100*t^13*y^6-106052943807668656200746780*t^12*y^7-\ 253029561778005601210693940*t^11*y^8-317930575749870019438063120*t^10*y^9-\ 232723437437811741780437952*t^9*y^10-117770785842413300676439950*t^8*y^11-\ 47963021169286346414223360*t^7*y^12-17236803380852668969359360*t^6*y^13-\ 5349432991326069153127680*t^5*y^14-1292226749133955644685824*t^4*y^15-\ 218012838567324670955520*t^3*y^16-22452260555685888000000*t^2*y^17-\ 1299829292236800000000*t*y^18+1374708088125000000*t^18+21139839043525000000*t^ 17*y-36972927797122700000*t^16*y^2-7637314643934949912000*t^15*y^3-\ 57809441678688932800000*t^14*y^4+1912447056858552924240212*t^13*y^5+ 43258939988044793464277560*t^12*y^6+103324958550406315972958000*t^11*y^7+ 135913779861602501149059380*t^10*y^8+100414721296593406498626090*t^9*y^9+ 49365463954780860644006814*t^8*y^10+18933281633035162546999680*t^7*y^11+ 6408722571815537843209200*t^6*y^12+1926073194020758672092160*t^5*y^13+ 458318634265167776020480*t^4*y^14+75684860944712450555904*t^3*y^15+ 7018283120025600000000*t^2*y^16+170565378048000000000*t*y^17-\ 4853730465050000000*t^17-6715279129892200000*t^16*y+1967320166737631900000*t^ 15*y^2+16343858189191125236000*t^14*y^3-602683443781520320413750*t^13*y^4-\ 14282309248740865601285712*t^12*y^5-33655469283904243196926480*t^11*y^6-\ 46907290353287117258899320*t^10*y^7-34976043864465433014562965*t^9*y^8-\ 16581519156552044396016720*t^8*y^9-5868957522096244375823592*t^7*y^10-\ 1802437498021247364834720*t^6*y^11-505356065329120659342080*t^5*y^12-\ 113405461799004428288000*t^4*y^13-17164587322880000000000*t^3*y^14-\ 1338955038720000000000*t^2*y^15+4684208671740200000*t^16-300555950885932400000 *t^15*y-3559466443815910250000*t^14*y^2+142342628297617716264000*t^13*y^3+ 3718678849432392938560000*t^12*y^4+8422579072009816830444904*t^11*y^5+ 12754992904100602951521480*t^10*y^6+9606743158645344025635520*t^9*y^7+ 4363034830429544913287125*t^8*y^8+1393104275196499428510000*t^7*y^9+ 369426318450498618896000*t^6*y^10+91669335693566103296000*t^5*y^11+ 18458097231188807680000*t^4*y^12+2309914240000000000000*t^3*y^13+ 116985856000000000000*t^2*y^14+20139419111021500000*t^15+498082805913583400000 *t^14*y-23664225944295160900000*t^13*y^2-733998670440203914088000*t^12*y^3-\ 1510936999776289723672500*t^11*y^4-2631465526919346238434000*t^10*y^5-\ 2006274397695498215642750*t^9*y^6-866838630683655897126250*t^8*y^7-\ 243754410757112872150000*t^7*y^8-51402134312682908800000*t^6*y^9-\ 10355544116975795200000*t^5*y^10-1759360000000000000000*t^4*y^11-\ 138444800000000000000*t^3*y^12-35050597917989800000*t^14+ 2499446312314518200000*t^13*y+103073411830196611700000*t^12*y^2+ 164234315332487304600000*t^11*y^3+387302170951188829825000*t^10*y^4+ 299719916738479888018750*t^9*y^5+122425955463889788125000*t^8*y^6+ 29547802256015355000000*t^7*y^7+4221982858960800000000*t^6*y^8+ 553523832776960000000*t^5*y^9+73728000000000000000*t^4*y^10-\ 125618046897160000000*t^13-9195434580513915000000*t^12*y-\ 3776230301524657500000*t^11*y^2-36291641280420331250000*t^10*y^3-\ 28518170718629288828125*t^9*y^4-10998769953415000000000*t^8*y^5-\ 2210470597920000000000*t^7*y^6-144572583168000000000*t^6*y^7+ 393637180681875000000*t^12-1468210987672812500000*t^11*y+ 1630464410067390625000*t^10*y^2+1296685108550000000000*t^9*y^3+ 475849108800000000000*t^8*y^4+76991316480000000000*t^7*y^5+ 143649722813593750000*t^11; t:= 1/2-1/2/u^3*(s+3)/(s^2+1)^2* (8*s^10+100*s^7-135*s^6+834*s^5-1205*s^4+2280*s^3-1365*s^2+890*s+321); y:=1/2+ 1/18*(28*s^9-235*s^8+556*s^7-1334*s^6+2174*s^5-3854*s^4+4360*s^3-4738*s^2+2362 *s-1047)*(s+3)/(s^2+1)/(s^6-7*s^4+42*s^3-45*s^2+34*s+7)/u; u2:=3*(8*s^2-13*s+17)*(s+3); ################################################ # icosahedral soln 46: # largest Valentiner solution # genus one, 24 branches th:=[1/3, 1/3, 1/3, 1/2]; y:=1/2 -1/2*(16*s^11+72*s^10+50*s^9-242*s^8-3143*s^7+6562*s^6-8312*s^5+9760*s ^4-9836*s^3+6216*s^2-2288*s+416)/(3*s^2-2*s+2)/(26*s^6+18*s^5-75*s^4+ 50*s^3+270*s^2-312*s+104)/u; t:=1/2+ 1/2*(s^2+4*s-2)*(8*s^10+16*s^9+24*s^8-84*s^7+429*s^6-312*s^5+258*s^4-\ 288*s^3+288*s^2-128*s+32)/(s+2)/(3*s^2-2*s+2)^2/u^3; u2:=(8*s^2-7*s+2)*(s+2); F1:= (95051008*t^12-570306048*t^11+1546384224*t^10-2504115680*t^9+7968020013*t^8-\ 23120752500*t^7+33266488974*t^6-23120752500*t^5+7968020013*t^4-2504115680*t^3+ 1546384224*t^2-570306048*t+95051008)*y^24+(-1140612096*t^12+6361196160*t^11-\ 15902990400*t^10-18670077864*t^9+103719204792*t^8-75507332688*t^7-75507332688* t^6+103719204792*t^5-18670077864*t^4-15902990400*t^3+6361196160*t^2-1140612096 *t)*y^23-68841472*t^22+(6502736832*t^12-33467942208*t^11+212402900052*t^10-\ 296673281784*t^9-323317666836*t^8+895340586096*t^7-323317666836*t^6-\ 296673281784*t^5+212402900052*t^4-33467942208*t^3+6502736832*t^2)*y^22+ 161144832*t^21+(256608128*t^13-25102247936*t^12-105417991816*t^11-518166275088 *t^10+2349584267808*t^9-1797345981192*t^8-1797345981192*t^7+2349584267808*t^6-\ 518166275088*t^5-105417991816*t^4-25102247936*t^3+256608128*t^2)*y^21-\ 166389760*t^20+(287712*t^14-2696399328*t^13+256645250070*t^12+934868782224*t^ 11-2016343174716*t^10-3462628594752*t^9+9590319708588*t^8-3462628594752*t^7-\ 2016343174716*t^6+934868782224*t^5+256645250070*t^4-2696399328*t^3+287712*t^2) *y^20+441914368*t^19+(786432*t^15-8775360*t^14-46131035256*t^13-1607715700272* t^12-185809165296*t^11+9243907858656*t^10-9424267990920*t^9-9424267990920*t^8+ 9243907858656*t^7-185809165296*t^6-1607715700272*t^5-46131035256*t^4-8775360*t ^3+786432*t^2)*y^19-640604928*t^18+(-7471104*t^15+489984292*t^14+520720708888* t^13+4079036415792*t^12-8045276687232*t^11-3345311057460*t^10+26374181686416*t ^9-3345311057460*t^8-8045276687232*t^7+4079036415792*t^6+520720708888*t^5+ 489984292*t^4-7471104*t^3)*y^18+441914368*t^17+(13958568*t^15-3928250448*t^14-\ 1724828158800*t^13-3533626382832*t^12+12097602162936*t^11-23284000365840*t^10-\ 23284000365840*t^9+12097602162936*t^8-3533626382832*t^7-1724828158800*t^6-\ 3928250448*t^5+13958568*t^4)*y^17-166389760*t^16+(5249349*t^16+29870532*t^15-\ 144081478530*t^14+2103092004204*t^13+4228726324185*t^12+7860035534040*t^11+ 41811644897208*t^10+7860035534040*t^9+4228726324185*t^8+2103092004204*t^7-\ 144081478530*t^6+29870532*t^5+5249349*t^4)*y^16+161144832*t^15+(32768*t^17-\ 42273320*t^16+1321414592*t^15+1227250220464*t^14-3226678787824*t^13-\ 22617684114168*t^12-37523953074528*t^11-37523953074528*t^10-22617684114168*t^9 -3226678787824*t^8+1227250220464*t^7+1321414592*t^6-42273320*t^5+32768*t^4)*y^ 15-68841472*t^14+(-245760*t^17-395420172*t^16-6410207952*t^15-2992544360976*t^ 14+14691677989776*t^13+55613932792572*t^12+51806838651072*t^11+55613932792572* t^10+14691677989776*t^9-2992544360976*t^8-6410207952*t^7-395420172*t^6-245760* t^5)*y^14+(14022192*t^17+3398284152*t^16-309743780112*t^15+1657821840144*t^14-\ 38242911924432*t^13-81739080098520*t^12-81739080098520*t^11-38242911924432*t^ 10+1657821840144*t^9-309743780112*t^8+3398284152*t^7+14022192*t^6)*y^13+(-\ 211040*t^18-85517528*t^17+12662109510*t^16+1888145059016*t^15+4605239973236*t^ 14+63190567069216*t^13+117639696624428*t^12+63190567069216*t^11+4605239973236* t^10+1888145059016*t^9+12662109510*t^8-85517528*t^7-211040*t^6)*y^12; F:=F1+(27648*t^ 19+1003584*t^18-3056189472*t^17-93264864120*t^16-4390711743360*t^15-\ 14185254856752*t^14-99958215034104*t^13-99958215034104*t^12-14185254856752*t^ 11-4390711743360*t^10-93264864120*t^9-3056189472*t^8+1003584*t^7+27648*t^6)*y^ 11+(-152064*t^19-5696928*t^18+17764022616*t^17-124407443484*t^16+7019419375104 *t^15+38135795921436*t^14+96322227692688*t^13+38135795921436*t^12+ 7019419375104*t^11-124407443484*t^10+17764022616*t^9-5696928*t^8-152064*t^7)*y ^10+(-348160*t^19+27408000*t^18+30790404528*t^17+787034182504*t^16-\ 16907449435224*t^15-46050188793664*t^14-46050188793664*t^13-16907449435224*t^ 12+787034182504*t^11+30790404528*t^10+27408000*t^9-348160*t^8)*y^9+(76032*t^20 +1946880*t^19-8652635856*t^18-196041886344*t^17+1511747453277*t^16+ 22292440117812*t^15+22708269761166*t^14+22292440117812*t^13+1511747453277*t^12 -196041886344*t^11-8652635856*t^10+1946880*t^9+76032*t^8)*y^8+(-304128*t^20+ 189057024*t^19+32609846016*t^18-376488363456*t^17-3891213440640*t^16-\ 12213863831232*t^15-12213863831232*t^14-3891213440640*t^13-376488363456*t^12+ 32609846016*t^11+189057024*t^10-304128*t^9)*y^7+(-18035200*t^20-484401152*t^19 +68827345024*t^18+1069871405288*t^17+1219870602400*t^16+8077351640048*t^15+ 1219870602400*t^14+1069871405288*t^13+68827345024*t^12-484401152*t^11-18035200 *t^10)*y^6+(110592*t^21+54008832*t^20-12132943872*t^19-197710177344*t^18-\ 208811321856*t^17-1601423698368*t^16-1601423698368*t^15-208811321856*t^14-\ 197710177344*t^13-12132943872*t^12+54008832*t^11+110592*t^10)*y^5+(-276480*t^ 21+579982080*t^20+24851278080*t^19-44938346928*t^18+613410271008*t^17-\ 177793804512*t^16+613410271008*t^15-44938346928*t^14+24851278080*t^13+ 579982080*t^12-276480*t^11)*y^4+(-46702592*t^21-756085760*t^20+34202770432*t^ 19-161154220800*t^18+31562618624*t^17+31562618624*t^16-161154220800*t^15+ 34202770432*t^14-756085760*t^13-46702592*t^12)*y^3+(70330368*t^21-9992277504*t ^20+28974397440*t^19-2533279872*t^18-6804262656*t^17-2533279872*t^16+ 28974397440*t^15-9992277504*t^14+70330368*t^13)*y^2+(1168760832*t^21-\ 2385702912*t^20-982213632*t^19+1058543616*t^18+1058543616*t^17-982213632*t^16-\ 2385702912*t^15+1168760832*t^14)*y; #more simply, if a:=3*y-t-1; b:=t^2+1-t; c:=2*t^3-3*t^2-3*t+2; #then (upto an overall constant): F:= 202262003*c^12+376297775168328*a^8*b^11*c^2+7677987796656*a^9*b^3*c^7-\ 61261544436264*a^9*b^6*c^5-19775383250112*a^9*b^9*c^3+368914430503296*a^9*b^12 *c+2502276659952*a^10*b*c^8+2799073309068*a^8*b^2*c^8+8060518163520*a^8*b^5*c^ 6-185864959840149*a^8*b^8*c^4+467323243917696*a^7*b^13*c-80373684738744*a^7*b^ 7*c^5-38333983090176*a^7*b^10*c^3-166307498842536*a^6*b^9*c^4+251178722134848* a^6*b^12*c^2+16997193431928*a^7*b^4*c^7+19615370749596*a^6*b^6*c^6-\ 747795992976*a^5*b^8*c^5-150693769068480*a^5*b^11*c^3+361783975454208*a^5*b^14 *c+5836390830936*a^5*b^5*c^7-83826046258704*a^4*b^10*c^4+64235607388704*a^4*b^ 13*c^2-478879689432*a^5*b^2*c^9+87330522864*a^4*b*c^10-2446097963814*a^4*b^4*c ^8+23360206913496*a^4*b^7*c^6-69185270088576*a^3*b^12*c^3+110683748353536*a^3* b^15*c+902847079008*a^3*b^6*c^7+11087512725312*a^3*b^9*c^5+9290015722368*a^2*b ^14*c^2-371171792520*a^3*b^3*c^9-16201849666752*a^2*b^11*c^4+5991899487264*a^2 *b^8*c^6+49712459052*a^2*b^2*c^10-901927930968*a^2*b^5*c^8+1777727627136*a*b^ 10*c^5-6691754186496*a*b^13*c^3+8453050472448*a*b^16*c-131345546112*a*b^7*c^7+ 1956048360*a*b*c^11-14561308368*a*b^4*c^9-1479189389052*a^14*b^2*c^6-\ 22941374138496*a^13*b^10*c+4411082380824*a^13*b^4*c^5-28138411509696*a^13*b^7* c^3+114400280*a^21*c^5-26787755990412*a^12*b^6*c^4-2162239264224*a^12*b^9*c^2-\ 839262493320*a^13*b*c^7+91857682931328*a^11*b^11*c+3560469953400*a^12*b^3*c^6-\ 17353237124160*a^11*b^8*c^3+3956780575632*a^11*b^2*c^7-24644324558232*a^11*b^5 *c^5-99930432*a^23*b^2*c^3-740890368*a^23*b^5*c+41060808*a^24*b^3*c^2-\ 7453263168*a^22*b^4*c^2-109356117120*a^20*b^5*c^2-16611873408*a^21*b^3*c^3-\ 31574530944*a^21*b^6*c+109518672*a^22*b*c^4-579828958848*a^19*b^7*c-\ 27197341116*a^20*b^2*c^4-26598600264*a^19*b*c^5-253654364736*a^19*b^4*c^3-\ 137430084672*a^20*b^8+365060151936*a^18*b^9+3281752853568*a^12*b^12-7214485*a^ 24*c^4+1826675712*a^22*b^7-4039439904*b^3*c^10-143862480*a^24*b^6-164557498180 *a^6*c^10+775343218176*b^12*c^4-207131403744*b^9*c^6+124985679433344*a^4*b^16+ 206246677244544*a^6*b^15+534534405896*a^9*c^9+166305067277616*a^8*b^14-\ 198951291238*a^12*c^8+103121485865856*a^10*b^13-8021537433216*a^14*b^11+ 22368596120*a^15*c^7-3339403363296*a^16*b^10-10586690740*a^18*c^6-\ 1732658273280*b^15*c^2+36719332992*b^6*c^8+25194045512*a^3*c^11+18549722774016 *a^2*b^17+1689228442368*b^18+1368312814008*a^6*b^3*c^8-516708993816*a^7*b*c^9-\ 1842528321792*a^18*b^6*c^2-137727037944*a^17*b^2*c^5-3505108562496*a^17*b^5*c^ 3-224700179328*a^17*b^8*c-3317808020688*a^16*b^7*c^2-3494092742910*a^16*b^4*c^ 4-18434297043072*a^15*b^9*c+40763428560*a^16*b*c^6-34509451092480*a^14*b^8*c^2 -2105768598696*a^15*b^3*c^5-2861195527872*a^15*b^6*c^3-319430258136*a^18*b^3*c ^4+3394256434776*a^14*b^5*c^4-79085089506024*a^10*b^7*c^4+155137969008000*a^10 *b^10*c^2-4458033038988*a^10*b^4*c^6; ####################################### ####################################### #For reference here are the first two #Dubrovin-Mazzocco icosahedral solutions #(1 was first found by Dubrovin in Springer LNM 1620), #then the two icosahedral solutions of Kitaev ################################### #DM icosahedral 1: th:=[0,0,0,-4/5]; P:=49-2133 *s^2+34308 *s^4-259044* s^6+1642878* s^8-7616646* s^10+ 13758708* s^12+5963724* s^14-719271 *s^16+42483* s^18; #nb: preprint version has typo: +16422878 s^8 y:=(s-1)^2*(1+3*s)^2*(-1+4* s+s^2)*(7-108* s^2+314* s^4-588 *s^6+119* s^8)^2/ ( (1+s)^3*(-1+3*s)* P); t:=( -1 + s ) ^5*( 1 + 3*s ) ^3*( -1 + 4*s + s^2 )/ ( ( 1 + s ) ^5*( -1 + 3*s ) ^3* ( -1 - 4*s + s^2 )); #################################### #DM icosahedral 2 th:=[0,0,0,-2/5]; P:= ( 9 - 342*s^2 + 4855*s^4 - 28852*s^6 + 63015*s^8 - 1942*s^10 + 121*s^12 ); y:=( -1 + s ) ^4*( 1 + 3*s ) ^2* ( -1 + 4*s + s^2 ) * ( 3 - 30*s^2 + 11*s^4 ) ^2/( ( 1 + s ) *( -1 + 3*s ) * ( 1 + 3*s^2 )* P); t:=( -1 + s ) ^5*( 1 + 3*s ) ^3* ( -1 + 4*s + s^2 )/( ( 1 + s ) ^5*( -1 + 3*s ) ^3* ( -1 - 4*s + s^2 )); ################################################## # Kitaev 3.4.2(1): #5 branches =icosahedral soln 21 th:=[1/5,1/5,2/5,2/5]; t:=(2*s^3*(s^2-5))/((s-2)^2*(s+3)^3); y:=(s^2*(s-1))/(3*(s-2)*(s+3)); F:=-75*y^2*t+45*y^3*t+50*y^3+ 15*y^4*t-60*y^4-15*y^2*t^2-9*y^5*t-5*y^3*t^2-t^2+45*y*t^2+18*y^5-8*t^3; #{ga = 2/25, al = 9/50, be = -1/50, de = 12/25} #puiseux expansion at zero has a branch with leading term t^(2/3)/5^(1/3) #and this is what we get from Jimbo's formula from the generators having #sigma values (1/3,1/2,1/3) #################################################### # icosahedral soln 26, from Kitaev's paper, after remark 2.7: th:=[1/5,1/5,1/5,1/3]; t:=(1/2-((800*s^4+960*s^3+312*s^2+100*s+15)*s)/ (2*(8*s+1)^2*u )); y:=(1/2-((40*s^3+22*s^2+16*s+3)*u)/ (2*(30*s^3+40*s^2+10*s+1)*(8*s+1))); u2:=(s*(8*s+1)*(5*s+4)); F:=(-178*t^3*y^9-948*t^3*y^7-543*t^3*y^8+24*t^3*y^5-243*t*y^8+54*y^9+543*t ^2*y^8-162*t*y^9+564*t^2*y^7-8*t^2*y^6-812*t^3*y^6+224*t^4*y^9+224*t^2*y^9-543*t^4*y^8+2712*t^4*y^7-1448*t^4 *y^6+2292*t^4*y^5-402*t^4*y^4-162*t^5*y^9+543*t^5*y^8-948*t^5*y^7-1448*t^5*y^6+2172*t^5*y^5-3000*t^5*y^4+764 *t^5*y^3+54*t^6*y^9-243*t^6*y^8+564*t^6*y^7-812*t^6*y^6+2292*t^6*y^5-3000*t^6*y^4+3008*t^6*y^3-972*t^6*y^2+ 54*t^6*y-8*t^7*y^6+24*t^7*y^5-402*t^7*y^4+764*t^7*y^3-972*t^7*y^2+378*t^7*y+54*t^8*y-27*t^7-27*t^8); #{be = -1/50, ga = 1/50, al = 2/9, de = 12/25} #check puiseux expansions: c1:=1/4*4^(4/5)*27^(1/5)*t^(4/5); #=that given by Jimbo's formula with sigmas=(1,2,1)/5 c2:=1/27*27^(2/3)*4^(1/3)*t^(2/3); #=that given by Jimbo's formula with sigmas=(1/3,1/3,1/5)