Elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic I am finding elliptic curves over $\mathbb C$ have same endomorphism ring but not isomorphic.
Elliptic curves over $\mathbb C$ can be identified with $\mathbb C/\land$ for some lattice $\land$. And endomorphism ring is just a set of z s.t. z$\land \subseteq \land$.
But I am stuck here, the lattice in my mind is too simple, I guess I need to see more example. Can somebody help me?
What about lattice generated by {$1$,$\sqrt{2} i$} and {$i$,$\sqrt{2} $}? Is this an example?
 A: Your two lattices actually give isomorphic elliptic curves, because one can be rotated into the other via multiplication by $i$.  So it doesn't work.
It is actually slightly tricky to find an example.  You need two lattices which are both closed under multiplication by some subring $R \subseteq \mathbb{C}$ strictly containing $\mathbb{Z}$, but are not multiples of each other.  Note that "multiple" can be any complex number.
One way to find an example is to fix a subring $R$, then look for sublattices within $R$ which are closed under multiplication by $R$.  By definition, these sublattices are the ideals of $R$.  So we are looking for 2 ideals of $R$ that are not multiples of each other by any complex number.
Now we enter the land of algebraic number theory: Two ideals of $R$ are complex multiples of each other iff they belong to the same ideal class.  So to find our example, we need a ring $R \subseteq \mathbb{C}$ with more than one ideal class.  This is equivalent to the statement that $R$ does not have unique factorization into elements.
The standard example is $R = \mathbb{Z}[\sqrt{-5}]$.  It has 2 ideal classes. There is the trivial ideal class represented by $R$ itself, and the non-trivial ideal class represented by the ideal $J = (2, 1 + \sqrt{-5})$.  So $R$ and $J$ represent two elliptic curves which are non-isomorphic, but both have endomorphism ring $R$.  For some more details on the ideal classes in $\mathbb{Z}[\sqrt{-5}]$, see here.
