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Small addition to "Higher genus icosahedral Painleve curves":
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On p.27 there is a remark that one can use the elliptic curve

(*)    u^2 = s^3 - 5*s^2 + 5*s 

rather than the elliptic curve 

(**)   z^2=(9*j^2-2*j+9)*(j^2-2*j+17) 

for the 20 branch genus one solutions (icosahedral solutions 44,45). The solutions are given as rational functions on (**), 
so in order to do this explicitly one needs a map between these two curves, such as:

j := -(3*s-u-2)/(s+u-2);    
z := 16*(2*s^2+u^2+2*u-5)/(s+u-2)^2;

Using this map the rational functions on (**) become rational functions on (*), as desired.

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For completeness, here are two other ways one can easily rewrite this elliptic curve:

As remarked in the paper, these elliptic curves are in the class 200B1 in Cremona's list. The minimal model in class 200B1 is:

(***) u^2 = s^3+s^2-3*s-2

(see for example:  https://www.lmfdb.org/EllipticCurve/Q/200b1/ ) and one can easily relate this to (*)
by shifting s by 2. The formulae to relate (**) and (***) are then:

j := (u-3*s-4)/(s+u);  
z := 16*(2*s^2+u^2+8*s+2*u+3)/(s+u)^2;

Using this, the rational functions on (**) giving the Painleve solutions become rational functions on (***). 

Finally, one could also put this elliptic curve in Weierstrass form:

(****)  u^2 = s^3 - 270*s - 675

Replacing (u,s) here by (27*u, 9*s-15) yields (*), and so it follows that the map:

j:=(u-9*s-81)/(u+3*s-9);
z:=16*(18*s^2+540*s+405+u^2+54*u)/(3*s+u-9)^2;

relates (**) and (****). Using this the rational functions on (**) giving the Painleve solutions become rational functions on (****). 

Note that all three of these maps are invertible.


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Note (as explained in Rmk 4) that all the solutions in the paper can be found electronically in the directory "solutions" in the gzip file here:

https://arxiv.org/e-print/math/0506407

This is the "source" for the arxiv files, linked to here: https://arxiv.org/format/math/0506407


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Maple code to check maps:
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restart;
j := -(3*s-u-2)/(s+u-2);    
z := 16*(2*s^2+u^2+2*u-5)/(s+u-2)^2;
c1:=z^2-(9*j^2-2*j+9)*(j^2-2*j+17); 
c2:=u^2 -( s^3 - 5*s^2 + 5*s);
simplify(c1/c2);   ## 2048/(s+u-2)^3

###########################

restart;
j := (u-3*s-4)/(s+u);  
z := 16*(2*s^2+u^2+8*s+2*u+3)/(s+u)^2;
c1:=z^2-(9*j^2-2*j+9)*(j^2-2*j+17); 
c2:=u^2 - (s^3+s^2-3*s-2);
simplify(c1/c2);   ## 2048/(s+u)^3

###########################

restart;
j:=(u-9*s-81)/(u+3*s-9);
z:=16*(18*s^2+540*s+405+u^2+54*u)/(3*s+u-9)^2;
c1:=z^2-(9*j^2-2*j+9)*(j^2-2*j+17); 
c2:=u^2 -(s^3 - 270*s - 675);
simplify(c1/c2);   ## 55296/(3*s-9+u)^3

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