curve #5056

y² = x³ − 297910322228830847139548024222961x

a-invariants: [0, 0, 0, -297910322228830847139548024222961, 0]
c-invariants: [14299695466983880662698305162702128, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 9533130311322587108465536775134752
discriminant (Δ): 1692141309090934248752193089070736499339342656265532089786496963378390689579378328252736975135339584
Faltings height: 17.7296
naive height: 235.9366
primes of bad reduction: 2, 3, 7, 13, 37, 41, 43, 47, 71, 127, 2287
regulator: 754395768.2723718960224471664625964240889361569012822
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:31 UTC
last updated: 2026-10-21 01:50:31 UTC

Witness: 7 independent points log in to add more points →

(5536145766238839024/25, 12986378870816463073220445732/125)
(590198031106914048, 453221852098920402517791408)
(2520269289388217803568496/61952641, 3623039662394469615405919115527723500/487629237311)
(-1801348406683558779770087/217149696, 4411305307806368373991431438290623747/3199917920256)
(-298749252018612589008806775/17567381764, 896099450701275193906046039669392417245/2328415913764088)
(72251309799151894845131100/124609, 614141699742771648011399057013299864070/43986977)
(-1846599421493957551239/10432900, 7737690092508460865430136413731259/33698267000)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 17260078859287719, rank 7 example from Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014, doi:10.5802/pmb.9; points from https://magma.maths.usyd.edu.au/~watkins/PTS.r6r7.

last edited by David Renshaw at 2026-10-21 01:50:31 UTC · history