Let $E \subset \mathbb{P}\_\mathbb{C}^2$ be an elliptic curve. If $E$ has complex multiplication (by anything) then the theory of complex multiplication in particular tells us that if $\sigma \in \textrm{Aut}(\mathbb{C})$ then $E^\sigma$ will be isogenous to $E$. Suppose $\textrm{End}(E) \cong \mathbb{Z}$ do we still have that $E^\sigma \cong E$ or are there some obvious counterexamples? What if we insist that the $j$-invariant, $j(E)$ is an algebraic number?



Isogeny classes of elliptic curves
We say that an elliptic curve $E$ over a number field $K$ is *an elliptic $\mathbf{Q}$-curve* if it is is isogenous to its Galois conjugates $E^\sigma$. These were first studied by Benedict Gross, but were later studied by Elkies, Ellenberg, Ribet, and many others. It's possible to show that $\mathbf{Q}$ curves are modular, independently of the BCDTW modularity theorem, and that even though they are defined over $K$, their $p$-torsion Galois representations (at least away from the degrees of said isogenies) descend down to $\mathbf{Q}$. Elkies also famously showed if $E\_{/K}$ is a $\mathbf{Q}$-curve without CM, there must be a geometrically isogenous curve $E'\_{/L}$ such that $Gal(L/\mathbf{Q}) \cong (\mathbf{Z}/2\mathbf{Z})^r$ for some integer $r$. This comes down to showing that the moduli space of elliptic $\mathbf{Q}$-curves of degree $N$ form a certain very special quotient of $X\_0(N)$. For details on most of this, see Jordan Ellenberg's "$\mathbf{Q}$-curves and Galois representations" ( <http://www.math.wisc.edu/~ellenber/MCAV.pdf> )


