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What is an example of an elliptic curve $E$ without complex multiplication? This means $End(E)=\mathbb{Z}$.

I know that complex elliptic curves are given by $\mathbb{C}^2/\Lambda$ for a lattice $\Lambda$, and if $\Lambda$ is a subring, like $\mathbb{Z} + \mathbb{Z}\sqrt{m}$ for $m<0$, then $\mathbb{C}^2/\Lambda$ has complex multiplication.

Also, I read that $j$ invariant of a CM elliptic curve is an algebraic integer, so the set of isomorphism classes of CM elliptic curves is countable, while the set of isomorphism classes of elliptic curves is $\mathbb{C}$, so in a sense most elliptic curves don't have complex multiplication.

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It is very easy to find an example. The curve $E\colon y^2=x^3+x+2$ has $j$-invariant $432/7$, which is not an alegbraic integer. Thus $E$ has no CM.

In general, if $\Lambda$ is of the form $\mathbb Z+\mathbb Z\tau$ for some $\tau\in \mathbb C$, then $\mathbb C/\Lambda$ has CM if and only if $\tau$ is imaginary quadratic. This makes it easy to construct tons of examples!

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  • $\begingroup$ I see, thanks. So taking $\tau=i\sqrt[3]{3}$ for example would give a non CM curve. $\endgroup$ – usr0192 Jan 11 '16 at 16:49

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