curve #4738

y² + y = x³ − 3175871497

a-invariants: [0, 0, 1, 0, -3175871497]
c-invariants: [0, 2743952973192]
rank (lower bound): ≥ 4
torsion subgroup: trivial
families: y² = x³ + k (Mordell curves); x³ + y³ = A (cube-sum curves)
conductor (N): 12703485987
discriminant (Δ): -4357221017991444832563
Faltings height: 2.8311
naive height: 57.2808
primes of bad reduction: 3, 109, 199
regulator: 127.3678004371098658330923167140705908563133408
submitted by: David Renshaw
submitted at: 2026-10-20 21:27:25 UTC
last updated: 2026-10-20 21:27:25 UTC

Witness: 4 independent points log in to add more points →

(2587, 118902)
(65073/4, 16593611/8)
(65073/16, 16203145/64)
(21691/9, 2808971/27)

Commentary

Cube-sum curve x^3 + y^3 = 21691 (Weierstrass model y^2 = x^3 - 432*21691^2). 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.

last edited by David Renshaw at 2026-10-20 21:27:25 UTC · history