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Say you're given an equation in the form $y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$. If the $a_i$'s are complex numbers, the subset $E^*\subset\mathbb{C}^2$ satisfying this equation is a punctured elliptic curve.

Is there a nice way to give an equation of the two loops $\gamma_1,\gamma_2 : S^1\rightarrow E^*$ (in terms of the $a_i$'s) which generate the fundamental group of $E^*$?

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    $\begingroup$ The Weierstrass $\wp$-function (searck Wikipedia for p-function) is the crucial ingredient here. Getting it to do what you want will require a little bit of work, though. $\endgroup$ – user64687 Nov 17 '14 at 8:40

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