TITLE: The Weil numbers and modulus of an elliptic curve
QUESTION [5 upvotes]: I have an ignorant question about elliptic curves which I'll be slightly imprecise about. If I have an elliptic curve $X$ defined over $\mathbb Z$, I can base change to $\mathbb C$, and then $X(\mathbb C)$ is isomorphic to the quotient of $\mathbb C$ by the lattice generated by $1, \tau$ for some $\tau \in \mathbb C$.
I can also base change to a finite field $\mathbb F_q$, and then the Frobenius operator on $H^1 (X( \bar {\mathbb{F} }_q), \bar {\mathbb{Q} }_l)$ has eigenvalues $\sigma, \bar \sigma$.  

Q: Is there any relationship between $\sigma, \bar \sigma$, and $\tau$? 

(I may have glossed over some things, like fixing an identification embedding of $\bar {\mathbb Q}$ into $\mathbb C$, or requiring $X(\mathbb F_q)$ to be smooth, or something else.)

REPLY [13 votes]: If $E$ does not have CM, then $\tau$ will be transcendental over $\mathbb Q$, so it's hard to imagine  a relation with the eigenvalues of Frobenius, which are integers in an imaginary quadratic field. However, if $E$ does have CM, then $\text{End}(E)\otimes\mathbb Q\cong\mathbb Q(\tau)$, and Frobenius lifts to an endomorphism of $E$ that is in the ring of integers of this field, so if you choose the right value for $\tau$, then they can be very closely related. If you want more details, read any standard treatment on complex multiplication and elliptic curves.