Tropical elliptic curves

[EnisKaya]

Consider the elliptic curve X/Q whose affine equation is given by

f(x,y) = y^2 - (x-6)(x-5)(x+11) = y^2 - x^3 + 91x - 330.

I'd like to consider this curve over Q_17, the field of 17-adic numbers, and draw the corresponding tropical curve and the dual subdivision. Here is my code (after homogenizing the curve and computing 17-adic valuation of the coefficients):

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$H = new Hypersurface<Min>(MONOMIALS=>[[0,2,1],[3,0,0],[1,0,2],[0,0,3]], COEFFICIENTS=>[0,0,0,0]); $ds = $H -> dual_subdivision(); $H -> VISUAL; $ds -> VISUAL;

When I visualize $ds, I see what I expect. However, when I visualize $H, what I see is a little bit "odd": the upward component is NOT vertical. Is this caused by 2D vs 3D?

In another direction, when I run the code

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print $H->GENUS;

the answer is 0. Shouldn't it be 1? From $ds, it's clear that the number of interior lattice points is 1.

I apologize in advance if these are silly questions. Polymake is highly new to me .

Remark: An elliptic curve is a smooth, projective, algebraic curve of genus one, with a distinguished point.

[joswig]

polymake's application tropical uses homogeneous coordinates, as you correctly figured out.

The pictures are drawn in a (skew) inhomogenous way (to save dimensions). The point with homogeneous coordinates (x,y,z) is equivalent to (0,y-x,z-x), and so it is drawn in the plane as (y-x,z-x). That dehomogenization is "skew" because it prefers one coordinate (the first one).

For more examples you might want to consult the computational Appendix C in my book https://bookstore.ams.org/gsm-219/.

[EnisKaya]

Thanks Michael! Now, I understand. In order to get the picture in my mind, the code should be

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$H = new Hypersurface<Min>(MONOMIALS=>[[1,0,2],[0,3,0],[2,1,0],[3,0,0]], COEFFICIENTS=>[0,0,0,0]); $H -> VISUAL;

where I'm homogenizing the curve with respect to the first coordinate.

Regarding the part

In another direction, when I run the code print $H->GENUS;, the answer is 0. Shouldn't it be 1? From $ds, it's clear that the number of interior lattice points is 1.

according to the documentation, the code " $H->GENUS; " returns the topological genus, which is obviously 0. I was expecting to see 1, since this is an elliptic curve. Sorry, my bad .