curve #5051

y² = x³ − 4272642074938259003328518537025x

a-invariants: [0, 0, 0, -4272642074938259003328518537025, 0]
c-invariants: [205086819597036432159768889777200, 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): 136724546398024288106512593184800
discriminant (Δ): 4991941792245878152480048261519832834225175914229232248692954732535076779123709929787641000000
Faltings height: 16.6685
naive height: 223.2030
primes of bad reduction: 2, 3, 5, 11, 17, 73, 179, 1297, 43481
regulator: 1531268431.219763467375827676714930498082693881088334
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:10 UTC
last updated: 2026-10-21 01:50:10 UTC

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

(41943099540089002835481/18386944, 3633283914844497926070439023144579/78843215872)
(71118835921918072561/144, 599754003200335557946340391241/1728)
(-78366968545187523501255/47416996, 16478767274363771280225220718322075/326513434456)
(1218790703104598450409/183184, 40443914526059155144260927459573/78402752)
(-427424277885894167698655/239444676, 163149888155500767095654296221069175/3705166916424)
(132064993737026798333820/8334769, 47583278079684180423271271196497550/24062478103)
(904822407628307710792678780/216950076841, 23639371101720281671989146766133465393250/101050789840924139)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 2067037027955295, 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 01:50:10 UTC · history