Singular

D.13.3.10 jInvariant

Usage:

jInvariant(f[,#]); f poly, # list

Assume:

- f is a a polynomial whose Newton polygon has precisely one interior lattice point, so that it defines an elliptic curve on the toric surface corresponding to the Newton polygon
- it the optional argument # is present the base field should be Q(t) and the optional argument should be one of the following strings:
'ord' : then the return value is of type integer, namely the order of the j-invariant
'split' : then the return value is a list of two polynomials, such that the quotient of these two is the j-invariant

Return:

poly, the j-invariant of the elliptic curve defined by poly

Note:

the characteristic of the base field should not be 2 or 3, unless the input is a plane cubic

LIB "tropical.lib";
ring r=(0,t),(x,y),dp;
// jInvariant computes the j-invariant of a cubic
jInvariant(x+y+x2y+y3+1/t*xy);
// if the ground field has one parameter t, then we can instead
//    compute the order of the j-invariant
jInvariant(x+y+x2y+y3+1/t*xy,"ord");
// one can compare the order of the j-invariant to the tropical j-invariant
tropicalJInvariant(x+y+x2y+y3+1/t*xy);
// the following curve is elliptic as well
poly h=x22y11+x19y10+x17y9+x16y9+x12y7+x9y6+x7y5+x2y3+x14y8;
// its j-invariant is
jInvariant(h);