I punti di coordinate razionali e, in particolare, di coordinate intere della cubica ellittica 1-011-011-01

Sansone, Giovanni

doi:10.1007/bf01789405

Annali di Matematica pura ed applicata

1980

DOI: 10.1007/bf01789405

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I punti di coordinate razionali e, in particolare, di coordinate intere della cubica ellittica 1-011-011-01

Giovanni Sansone¹

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1983

1986

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“…(This is the curve denoted 92c in the tables of Birch and Kuyk [3]; see also the review [MR 82g: 10037] by the first author of a paper by Sansone [16].) Recently, Mestre [14] has given an example of a curve of rank at least 12 with at least 320 integer points.…”

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Integer points on 𝑦²=𝑥³-7𝑥+10

Bremner

Tzanakis

1983

Math. Comp.

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It is apparent that the groups of rational points on curves (0.1)-(0.4) will probably have several generators. Using the arithmetic of the field Q(p), p3 -7p + 10 = 0 (see Section 1), one can perform the standard 2-descents (see Cassels [6] or Birch and Swinnerton-Dyer [4]) on the curve (0.1) to determine that its rational Mordell-Weil group actually has rank 2, but the details are omitted. It seems plausible that the rank of the curves (0.2), (0.3), (0.4) is in each instance equal to 4, but this has not been specifically verified.It is perhaps also worth noting here the integer point on (0.4) given by (x, y) = (1645085185,66724078854865) because of the large size of the coordinates; Lang [11], [12] has made some interesting conjectures on the size of integral points on the curve y2 = x3 + ax + b, in particular that for an integral point (x0, y0) then |x0|« max(|a|3,|èp)* for some uniform k. The curve (0.1) has conductor 23 • 83 (see Täte [19] for a recipe on calculation of the conductor), rank 2, and 26 integer points, none of which, however, is particularly Targe'. Examples possessing a sizeable solution include:(i) the curve j2 = x3 -x + 1 of conductor 22 • 23, rank 1, and 12 integer points including (x, y) = (56,419). (This is the curve denoted 92c in the tables of Birch and Kuyk [3]; see also the review [MR 82g: 10037] by the first author of a paper by Sansone[16].)

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Integer points on 𝑦²=𝑥³-7𝑥+10

Bremner

Tzanakis

1983

Math. Comp.

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Some questions in the geometric theory of invariants of groups generated by orthogonal and oblique reflections

Ignatenko¹

1986

J Math Sci

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I punti di coordinate razionali e, in particolare, di coordinate intere della cubica ellittica 1-011-011-01

Cited by 2 publications

References 1 publication

Integer points on 𝑦²=𝑥³-7𝑥+10

Integer points on 𝑦²=𝑥³-7𝑥+10

Some questions in the geometric theory of invariants of groups generated by orthogonal and oblique reflections

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