curve #3698

y² + xy = x³ + x² − 9707860480137448101821815861x + 365111485439458059801777678001548174488125

a-invariants: [1, 1, 0, -9707860480137448101821815861, 365111485439458059801777678001548174488125]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 1321043258541830079458759813032686
discriminant (Δ): 964825460567005894728668892146784215021796339926265191062199141002557012276667124784
Faltings height: 14.9355
naive height: 204.9418
primes of bad reduction: 2, 3, 7, 13, 17, 43, 101, 131, 179, 509, 587, 1367, 2141, 13217, 120917
regulator: 243956883642.2575329775027833404935389483320418026183786484
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:45:27 UTC
last updated: 2026-09-30 01:45:27 UTC

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

(61435597409838, 24117538879733583959)
(-490333896749952107302097/21789892996, 2433018901538236122147388880137627577/3216493264711544)
(8317707424784025/121, 200131788918325932684635/1331)
(17649623872748926/289, 20711172715370776160939/4913)
(8305391177311539257/179776, 9405140484931026536195017505/76225024)
(5340555814390784184878/32982049, 330185185862186825116505189040281/189415907407)
(2227846904064755674130350/36239594689, 175396223753844719113781864380281545/6898822922160863)
(61207878462315758738739094/145517072089, 466746429128827780582666251321021545501/55509960938574563)
(201012838324927296239137394775/3006045325927396, 20159072484016894271671227828055849024250465/164813691587386274433944)
(1036103050555591404471/14160169, -11493951401532414889885430840059/53284715947)

Commentary

Specialization at t = 4455/18784 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), 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 2026-09-30 01:45:27 UTC · history