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D.13.6.10 jInvariant
Procedure from library tropical.lib (see tropical_lib).
- 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
Example:
| | 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
jInvariant(x+y+x2y+y3+1/t*xy);
==> (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
jInvariant(x+y+x2y+y3+1/t*xy,"ord");
==> -6
// one can compare the order of the j-invariant to the tropical j-invariant
tropicalJInvariant(x+y+x2y+y3+1/t*xy);
==> // ** 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
jInvariant(h);
==> 912673/528
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