Singular

D.13.2.17 ellipticNFDB

Usage:

ellipticNFDB(n[,#]); n int, # list

Assume:

n is an integer between 1 and 16

Purpose:

this is a database storing the 16 normal forms of planar polygons with precisely one interior point up to unimodular affine transformations
(see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons and the number 12. Amer. Math. Monthly 107 (2000), no. 3, 238--250.)

Return:

list, L such that
L[1] : list whose entries are the vertices of the nth normal form
L[2] : list whose entries are all the lattice points of the nth normal form
L[3] : only present if the optional parameter # is present, and then it is a polynomial in the variables (x,y) whose
Newton polygon is the nth normal form

Note:

the optional parameter is only allowed if the basering has the variables x and y

LIB "polymake.lib";
==> Welcome to polymake version
==> Copyright (c) 1997-2015
==> Ewgenij Gawrilow, Michael Joswig (TU Darmstadt)
==> http://www.polymake.org
list nf=ellipticNFDB(5);
// the vertices of the 5th normal form are
nf[1];
==> [1]:
==>    3,0
==> [2]:
==>    0,0
==> [3]:
==>    0,3
// its lattice points are
nf[2];
==> [1]:
==>    3,0
==> [2]:
==>    2,1
==> [3]:
==>    1,2
==> [4]:
==>    0,3
==> [5]:
==>    2,0
==> [6]:
==>    0,2
==> [7]:
==>    1,0
==> [8]:
==>    0,1
==> [9]:
==>    0,0