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We perform descent calculations for the families of elliptic curves whose m-torsion splits as µm × Z/mZ for m = 3, 4 or 5. These curves are parametrised by the modular curve X(m) = P1, whose cusps are arranged as the vertices of one of the Platonic solids. Following McCallum [McC] we write the Cassels-Tate pairing as a sum of local pairings. In the case m = 5 our results extend the work of Beaver [Be].
The paper ends with some numerical examples. In the case m = 5 we were unable to find the expected number of generators on all the elliptic curves considered, owing to these points having large height. All the remaining generators were computed by Mark Watkins (using four descent), and sent to me in September 2003. The completed versions of Tables 3 and 4 ( = Tables 3.3 and 3.4 in the JNT article) are here. In particular this data includes the generator for the rank 1 curve considered in [Be].