curve #3670

y² + xy = x³ − 228454473893195280390x + 1329069192073918706341388999475

a-invariants: [1, 0, 0, -228454473893195280390, 1329069192073918706341388999475]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 17261904761372158878705
discriminant (Δ): 82689873973912076822261851698845658130683916014781640625
Faltings height: 10.1900
naive height: 152.2472
primes of bad reduction: 3, 5, 11, 13, 19, 43, 71, 151, 163, 337, 739, 1741
regulator: 5361328.48060811591303773371689646505187299500890508506
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:30:18 UTC
last updated: 2026-09-29 19:30:18 UTC

Witness: 8 independent points log in to add more points →

(8721670755, 726343612185)
(95241484965/16, 27159731366842965/64)
(138245827125/16, 889862391103485/64)
(349252042905/64, 253367491715495265/512)
(424592674230/49, 3396701932025835/343)
(429420245745/49, 2062217613341955/343)
(17569412239830/841, 58095097820577529965/24389)
(2970570585278505/341056, 530017744171057259865/199176704)

Commentary

Specialization at t = 19/302 of the Elkies-Klagsbrun Z/4 fibration E2: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = -8(80t+9), c = 2(t-2)(2t-1)(18t-1)(2t-81) (arXiv:2003.00077, eq. (6)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:30:18 UTC · history