curve #3674

y² + xy + y = x³ − x² − 73618642017474515398636907x + 243125105184756200058212321227222468539

a-invariants: [1, -1, 1, -73618642017474515398636907, 243125105184756200058212321227222468539]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 2108987770780492075994930190
discriminant (Δ): 1198338420457818027531344205257713365564302642421258048393604380386918400
Faltings height: 13.3433
naive height: 190.2964
primes of bad reduction: 2, 3, 5, 11, 31, 37, 67, 101, 137, 181, 191, 199, 331, 929, 947
regulator: 9541726373.4946771750324308799045454457925516233777872084
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:30:59 UTC
last updated: 2026-09-29 19:30:59 UTC

Witness: 9 independent points log in to add more points →

(698851624325627/121, 4332202408810337707686/1331)
(6299370158046971/3025, 1654151718351078793755654/166375)
(6758551796688371/1369, 3293419745959078534914/50653)
(7755229551031763/2809, 1161994554080872655119842/148877)
(11898383005383659/2401, 792291148396733257098/117649)
(52580899754606331/9409, 2280010624034622723221546/912673)
(288354786617579507/80089, 112732964601982151559221886/22665187)
(1647140965456744091/69169, 1992264742064149198734720486/18191447)
(518033244135787288680203/104634781729, 365569647016359161175931777278282/33846526750224817)

Commentary

Specialization at t = 905/402 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 19:30:59 UTC · history