We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large.
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Tom Fisher, Some examples of 5 and 7 descent for elliptic curves over Q. J. Eur. Math. Soc. 3 (2001), no. 2, pp. 169–201DOI 10.1007/S100970100030