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I know that every such surface is equivalent to some torus. So the question is how many non equivalent tori are there, and can we more or less easily figure it out.

Big Coconut
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    $j$-invariant, and it is torus/tori, not thorus/thoruses. – user10354138 Sep 10 '23 at 15:10
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    In more geometric (less analytic number-theoretic) terms, a genus-one Riemann surface intrinsically corresponds to a plane lattice up to rotation and real scaling (i.e., up to complex scaling). (Given a lattice, the quotient of the plane by that lattice is a torus with holomorphic structure.) Customarily, one assumes the lattice is generated by $1$ and a complex number $\tau$, which without loss of generality may be assumed to have positive imaginary part, magnitude at least $1$, and absolute real part no larger than $1/2$. – Andrew D. Hwang Sep 10 '23 at 15:58
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    Does this answer your question? When are two tori biholomorphic? (Suggesting a closed question because its accepted answer looks close to the spirit of this question.) – Andrew D. Hwang Sep 10 '23 at 16:00
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    @AndrewD.Hwang,yes, I'm reading it right now! – Big Coconut Sep 10 '23 at 16:01

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