Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic\n norm equations

We explain a method for computing the Cassels-Tate pairing on the 3-isogeny\nSelmer groups of an elliptic curve. This improves the upper bound on the rank\nof the elliptic curve coming from a descent by 3-isogeny, to that coming from a\nfull 3-descent. One ingredient of our work is a new algorithm for solving cubic\nnorm equations, that avoids the need for any S-unit computations. As an\napplication, we show that the elliptic curves with torsion subgroup of order 3\nand rank at least 13, found by Eroshkin, have rank exactly 13.\n