I was reading Silverman's Advanced Topics in the Arithmetic of Elliptic Curves book and I saw a notation which it wasn't defined before (or I missed):
Let $E$ be an elliptic curve over $\mathbf{C}$ and $\sigma: \mathbf{C} \longrightarrow \mathbf{C}$ be a field automorphism. Silverman writes $E^{\sigma}$ at page 104. Now what is exactly $E^{\sigma}$? Let $E$ be given by the Weierstaß equation
$$Y^2Z + a_1 XYZ + a_3YZ^2 = X^3 + a_2X^2Z + a_4XZ^2 + a_6Z^3$$ Then is the curve $E^{\sigma}$ given by
$$Y^2Z + \sigma(a_1) XYZ + \sigma(a_3)YZ^2 = X^3 + \sigma(a_2)X^2Z + \sigma(a_4)XZ^2 + \sigma(a_6)Z^3?$$
Or does $E^{\sigma}$ mean something else?
Thanks!