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It is well known that isomorphism class of elliptic curve and lattice up to homeothetic corresponds bijectively.

But I don't have concrete examples. Can we figure out lattice from given weierstrass equation?

For example,

What is a lattice of elliptic curve $y^2＝x^3-x$ ?

Pont
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    Related: https://math.stackexchange.com/questions/766229, https://math.stackexchange.com/questions/1291245, https://math.stackexchange.com/questions/309882/. I guess the lattice should be $\Bbb Z[i]$. Note that your curve has CM by this order in $\Bbb Q(i)$. – Watson Jun 07 '21 at 16:49
  • note that (1,0), (0,0), and (-1,0) are part of the lattice - am I interpreting your question correctly? – Moti Jun 07 '21 at 18:56
  • Also note that x selection is limited to be "far" from prime. – Moti Jun 07 '21 at 18:58
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    @Moti, you have noted points on the curve $E$. However, OP is asking not about this - there is a theorem that an elliptic curve over $\mathbb{C}$ has a complex analytic group isomorphism with the Riemann surface $\mathbb{C}/\Lambda$ for some rank $2$ $\mathbb{Z}$-module $\Lambda \subset \mathbb{C}$; OP wants to know what $\Lambda$ is. – Mummy the turkey Jun 07 '21 at 20:46

1 Answers1

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The curve you asked about ($y^2 = x^3 - x$) is LMFDB 32.a3. Here the Weierstrass invariants are $\,g_2=-1/4,\, g_3=0\,$ and this is known as the pseudo-lemniscate case. See DLMF 23.5.iv for some details. The lattice for this curve is a square lattice with side length $$\Gamma(1/4)^2/\sqrt{2\pi}=5.244115108584239620\dots$$ and the period parallelogram is a square one of whose diagonals is along the real axis.

Somos
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