Can someone give me an example of elliptic curve with CM by sqrt(-7) with the action. I've found a list of examples in the following link but not the action.
http://planetmath.org/examplesofellipticcurveswithcomplexmultiplication
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Sign up to join this communityCan someone give me an example of elliptic curve with CM by sqrt(-7) with the action. I've found a list of examples in the following link but not the action.
http://planetmath.org/examplesofellipticcurveswithcomplexmultiplication
I am sure others (magma, etc) can do the same. With sage the following lines will produce the rational functions $\bigl(f(x,y), g(x,y)\bigr)$ representing the multiplication by $(1\pm\sqrt{-7})/2$ on $E$ in the Weierstrass equation given at the link. Once you have that you can get all endomorphisms.
sage: E = EllipticCurve([1,-1,0,-2,-1])
sage: K.<t> = NumberField(x^2+7)
sage: EK = E.base_extend(K)
sage: psis = EK.isogenies_prime_degree(2)
sage: [psi.codomain().is_isomorphic(EK) for psi in psis]
[True, True, False]
sage: psi = psis[0]
sage: iota = psi.codomain().isomorphism_to(EK)
sage: psi.set_post_isomorphism(iota)
sage: psi.rational_maps()
whose $x$-coordinate $f(x,y)$ is $$\frac{(\sqrt{-7} - 3)\cdot x^2 + (-2\, \sqrt{-7} - 2)}{8\,x + \sqrt{-7} + 5}$$
See J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves (Springer GTM), Proposition 2.3.1. Note that since the class number of $\mathbb{Q}(\sqrt{-7})$ is 1, this is the only example defined over $\mathbb{Q}$.