Journal of the European Mathematical Society
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Some examples of 5 and 7 descent for elliptic curves over QTom Fisher (1) DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB3 0WB, CAMBRIDGE, UNITED KINGDOM
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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Fisher Tom: Some examples of 5 and 7 descent for elliptic curves over Q. J. Eur. Math. Soc. 3 (2001), 169-201. doi: 10.1007/s100970100030