Computing the Cassels–Tate pairing on 3-isogeny Selmer groups via cubic norm equations
We explain a method for computing the Cassels–Tate pairing on the $3$-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by $3$-isogeny, to that coming from a full $3$-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any $S$-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order $3$ and rank at least $13$, found by Eroshkin, have rank exactly $13$.