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D.13.4.10 jInvariant

Procedure from library tropical.lib (see tropical_lib).

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

- 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

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

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

LIB "tropical.lib";
==> Welcome to polymake version
==> Copyright (c) 1997-2015
==> Ewgenij Gawrilow, Michael Joswig (TU Darmstadt)
==> http://www.polymake.org
ring r=(0,t),(x,y),dp;
// jInvariant computes the j-invariant of a cubic
==> (32768t12+24576t10+3072t8-1024t6-96t4+24t2-1)/(256t12+32t10+11t8-t6)
// if the ground field has one parameter t, then we can instead
//    compute the order of the j-invariant
==> -6
// one can compare the order of the j-invariant to the tropical j-invariant
==> // ** int division with `/`: use `div` instead in line >>  genus=-genus/2\
   ; // we have counted each bounded edge twice<<
==> 6
// the following curve is elliptic as well
poly h=x22y11+x19y10+x17y9+x16y9+x12y7+x9y6+x7y5+x2y3+x14y8;
// its j-invariant is
==> 912673/528