Singular

D.13.3.9 weierstrassForm

Usage:

weierstrassForm(wf[,#]); wf poly, # list

Assume:

wf 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

Return:

poly, the Weierstrass normal form of the polynomial

Note:

- the algorithm for the coefficients of the Weierstrass form is due to Fernando Rodriguez Villegas, villegas@math.utexas.edu
- the characteristic of the base field should not be 2 or 3
- if an additional argument # is given, a simplified Weierstrass form is computed

LIB "tropical.lib";
ring r=(0,t),(x,y),lp;
// f is already in Weierstrass form
poly f=y2+yx+3y-x3-2x2-4x-6;
weierstrassForm(f);
// g is not, but wg is
poly g=x+y+x2y+xy2+1/t*xy;
poly wg=weierstrassForm(g);
wg;
// but it is not yet simple, since it still has an xy-term, unlike swg
poly swg=weierstrassForm(g,1);
swg;
// the j-invariants of all three polynomials coincide
jInvariant(g);
jInvariant(wg);
jInvariant(swg);
// the following curve is elliptic as well
poly h=x22y11+x19y10+x17y9+x16y9+x12y7+x9y6+x7y5+x2y3;
// its Weierstrass form is
weierstrassForm(h);