A309164 -id:A309164 - cp's OEIS frontend

A309162 Rank of elliptic curve y^2 = x * (x+1) * (x+n).

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 2, 0, 1, 1, 0, 2, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1

Offset: 2

Seiichi Manyama, Jul 15 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 2..1000
H. Mishima, Tables of Elliptic Curves

Cf, A306929, A309163, A309164.

{a(n) = ellanalyticrank(ellinit([0, n+1, 0, n, 0]))[1]}

A372543 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x + 1 is n, or -1 if no such k exists.

Original entry on oeis.org

0, 1, 2, 4, 8, 17, 61, 347, 3778, 11416

Offset: 0

Jose Aranda, Jul 04 2024

This family of curves quickly reaches a moderate value of rank with a relatively small parameter k.

By heuristic search (see links), a(10) <= 216493 and a(11) <= 1448203.

Anna Antoniewicz, On a family of elliptic curves, (2005) Iagellonicae Acta Mathematica, XLIII.
Jose Aranda, Non sequential search for upper bounds (PARI-GP Script)
Cecylia Bocovich, Elliptic Curves of High Rank, (2012) Macalester College, Science Honors Projects.

Cf. A194687, A309060, A309068, A309069, A309146, A309147, A309164, A309165.

a(n,startAt=0)=for(k=startAt, oo, my(t=ellrank(ellinit([-k^2, +1]))); if(t[2]n, warning("k=",k," has rank in ",t[1..2]); next); if(t[1]n, error("Cannot determine if a(",n,") is ",k," or larger; rank is in ",t[1..2])); return(k)) \\ Charles R Greathouse IV, Jul 08 2024

PARI
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\\ See Aranda link.
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A309162 Rank of elliptic curve y^2 = x * (x+1) * (x+n).

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A372543 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x + 1 is n, or -1 if no such k exists.

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