Actually, we don’t need to wait for anymore help. A little bit of playing around with the projective data ends up revealing the special projection where the elliptic period integral obeys a particularly nice ode:
In[]:=
data={3(x-3y)(81x^3-9x^2y-53xy^2-27y^3),1(x-3y)*(3^6x^5-3^5*5*x^4y-2*3^35^2x^3y^2-350*x^2*y^3-3^3*5^2xy^4-243*y^5)};
In[]:=
specialTransform={y->(1/3+1/3x),x->x-1,x->27x/32};
In[]:=
specialData=Factor[Fold[ReplaceAll,data,specialTransform]];
In[]:=
ode=-EllipticPeriodsAnnihilator[pfcs1]@@specialData
Out[]=
{5(-16+21x),4(32-118x+81),4(-1+x)x(-32+27x)}
2
x
In[]:=
previous=Factor[{5(21*x-16),D[#,x],#}&[4*x*(x-1)*(27*x-32)]];
In[]:=
ode-previous
Out[]=
{0,0,0}
In[]:=
FrobeniusTerms[Evaluate[ode],3,16]
Out[]=
{1,10,120,1540,20500,279480,3876600,54496200,774468900,11107261000,160553895040}
In[]:=
FrobeniusTerms[Evaluate[ode/.x->x+32/27],3,-16]
Out[]=
{1,30,1440,85260,5606100,391231080,28360117800,2110794125400,160187289344100,12339496371120600,961855480344860640}
conclusion
conclusion
Quite amazingly, we’ve now proven that periods of the Icosahedral Goursat surface are elliptic integrals by giving explicit g-data how to write the integrand in Weierstrass normal form. Before entering anything to OEIS, some thought needs to be given toward how to choose “special projections” and obtain nice, minimal ODEs for the period annihilators.