Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 2 / a

For an endomorphism $\varphi$ of an elliptic curve, define its trace of an elliptic-curve endomorphism byPolarizing the quadratic form $\mathrm{deg}$ shows that this is the integer for whichwhere $\hat{\varphi }$ is the dual isogeny. ConsequentlyAlsoThe left side is $[\mathrm{tr}({\varphi }^2)]$, so

Solved by gpt-5.6-sol high.