TITLE: How to get explicit unramified covers of an elliptic curve?
QUESTION [11 upvotes]: Say we begin with an explicit elliptic curve over $\mathbb{C}$, say: $y^2=x(x-1)(x-2)$. According to abstract reasoning this elliptic curve has an (several, in fact) unramified cover with group $C_n$. My question is, how can one find the equations that define this cover? To be very precise: how can one describe the function field extension of $Quot(\mathbb{C}[x,y]/(y^2-x(x-1)(x-2)))$ as the quotient field of a polynomial ring modulo an explicit ideal?
It would be nice to have a method that can generate all such unramified $C_n$ covers, but a method that generates some unramified $C_n$ cover would be helpful too.

REPLY [6 votes]: Let me expand Felipe's answer a bit.
Vélu's formulae given in the very readable short paper [1]  are very easy to use. For instance if you are given a $n$-torsion point one can immediately write down the Weierstrass equation for the quotient by this point. 
To get such a point, one can use the division polynomials which can be computed recursively very fast. The zeroes of this polynomial are exactly the $x$-coordinates of the $n$-torsion points. Oover the complex number it is not difficult to get them also using the parametrisation by the Weierstrass $\wp$-function.
There are quite a few paper improving Vélu's formula. Most importantly, there is Kohel's thesis [2]. Or for instance [3]. These were used in the implementation for isogenies in sage [sage] and in [magma]. 
[1] MR0294345, Jacques Vélu, Isogénies entre courbes elliptiques. C. R. Acad. Sci. Paris Sér. A-B 273, 1971.
[2] MR2695524, David Kohel, Endomorphism rings of elliptic curves over finite fields. Thesis (Ph.D.), University of California, Berkeley. 1996.
[3] MR2398793, Bostan, A.; Morain, F.; Salvy, B.; Schost, É. Fast algorithms for computing isogenies between elliptic curves. Math. Comp. 77 (2008).
[magma] http://magma.maths.usyd.edu.au/magma/handbook/text/1321
[sage] http://www.sagemath.org/doc/reference/sage/schemes/elliptic_curves/ell_curve_isogeny.html