I have to prove that $\Bbb C/(1+ib)\Bbb Z$ is an elliptic curve, where $b\in\{1,2,3\}$. The problem is that the teacher was a bit broad giving the definition of elliptic curve. He presented them as tori like $\Bbb C/\Lambda$ where $\Lambda=\omega_1\Bbb Z+\omega_2\Bbb Z$ is a lattice.
Now,$(1+ib)\Bbb Z$ is not a lattice and the space $\Bbb C/(1+ib)\Bbb Z$ can be seen as $\{0\le\Im z<b\}$ indetifying the upper and lower lines $\Im z=0$ and $\Im z=b$ respectively, getting an infinite cylinder. Even identifying the point at infinity, it doesn't sound good.
Hints?