An interesting family of curves of genus 1

Andrew Bremner

International Journal of Mathematics and Mathematical Sciences, 2000, vol. 23, 1-4

Abstract:

We study the family of elliptic curves y 2 = x 3 − t 2 x + 1 , both over ℚ ( t ) and over ℚ . In the former case, all integral solutions are determined; in the latter case, computation in the range 1 ≤ t ≤ 999 shows large ranks are common, giving a particularly simple example of curves which (admittedly over a small range) apparently contradict the once held belief that the rank under specialization will tend to have minimal rank consistent with the parity predicted by the Selmer conjecture.

Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:362171

DOI: 10.1155/S0161171200002210

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