curve #4747

y² = x³ − 126703627313158216226427307363854675

a-invariants: [0, 0, 0, 0, -126703627313158216226427307363854675]
c-invariants: [0, 109471933998568698819633193562370439200]
rank (lower bound): ≥ 10
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 506814509252632864905709229455418700 ☆ record for cube-sum curves, rank ≥ 10
discriminant (Δ): -6935245563302651263483549978476274664745938495668130773655811392361630000 ☆ record for cube-sum curves, rank ≥ 10
Faltings height: 12.6558 ☆ record for cube-sum curves, rank ≥ 10
naive height: 175.1775 ☆ record for cube-sum curves, rank ≥ 10
primes of bad reduction: 2, 3, 5, 7, 23, 31, 37, 43, 83, 109, 151, 421
regulator: 38376357796748.64469403093719142970103873967896154480080125
submitted by: David Renshaw
submitted at: 2026-10-20 21:28:01 UTC
last updated: 2026-10-20 21:28:01 UTC

Witness: 10 independent points log in to add more points →

(3528379374409/4, 5984763094833906277/8)
(6110958239731/9, 11655021356694595396/27)
(10497983708011, 34012226270126447216)
(83919846442489/36, 764916625222063960787/216)
(198185567469961/36, 2788965029861620279109/216)
(315390907055911/289, -5321096238055550334866/4913)
(249889953577441/484, 1112982603584253549839/10648)
(8774707419959359/1225, 821815055330484805887898/42875)
(10442897018676361/2916, -1065691846252244332562491/157464)
(5384554912082001511/3214849, 12325049054083396213643921734/5764224257)

Commentary

Cube-sum curve x^3 + y^3 = 137006962414679910 (Weierstrass model y^2 = x^3 - 432*137006962414679910^2). Source: Elkies & Rogers, 'Elliptic Curves x^3+y^3=k of High Rank', ANTS-VI (LNCS 3076, 2004), arXiv:math/0403116, https://arxiv.org/abs/math/0403116 Points mapped from the 3-isogenous curve uv(u+v)=A given there via the dual isogeny.

last edited by David Renshaw at 2026-10-20 21:28:01 UTC · history