The cyclic twist of elliptic curve is a principally polarized abelian variety

Question

Let $L/K$ is a cyclic extension of degree $p$, and let $E/K$ be an elliptic curve.

Let $E^L$ be the kernel of the map $Res^L_{K}(E) \rightarrow E$, where $Res^L_{K}(E)$ is the Weil-restriction.

Is the twist $E^L$ principally polarized?

Answer 1

Usually, $E^L$ is not principally polarised. See E. Howe, Isogeny Classes of Abelian Varieties with no Principal Polarizations, where it is shown that under some mild hypotheses every polarisation of $E^L$ has degree divisible by $p^2$.