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D.13.6.9 weierstrassForm
Procedure from library tropical.lib (see tropical_lib).
- 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
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),lp;
// f is already in Weierstrass form
poly f=y2+yx+3y-x3-2x2-4x-6;
weierstrassForm(f);
==> -x3-2*x2+xy-4*x+y2+3*y-6
// g is not, but wg is
poly g=x+y+x2y+xy2+1/t*xy;
poly wg=weierstrassForm(g);
wg;
==> -x3+2*x2+1/(t)*xy-x+y2
// but it is not yet simple, since it still has an xy-term, unlike swg
poly swg=weierstrassForm(g,1);
swg;
==> -x3+(8t2-1)/(4t2)*x2-x+y2
// the j-invariants of all three polynomials coincide
jInvariant(g);
==> (-4096t12+12288t10-13056t8+5632t6-816t4+48t2-1)/(16t10-t8)
jInvariant(wg);
==> (-4096t12+12288t10-13056t8+5632t6-816t4+48t2-1)/(16t10-t8)
jInvariant(swg);
==> (-4096t12+12288t10-13056t8+5632t6-816t4+48t2-1)/(16t10-t8)
// the following curve is elliptic as well
poly h=x22y11+x19y10+x17y9+x16y9+x12y7+x9y6+x7y5+x2y3;
// its Weierstrass form is
weierstrassForm(h);
==> -x3+1728*x+y2+27648
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