curve #3628

y² + xy = x³ − 13799859692500506680x + 19254614592101304671643180096

a-invariants: [1, 0, 0, -13799859692500506680, 19254614592101304671643180096]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 2681265063185100898944986344566
discriminant (Δ): 8031718667234367655099251283166268195377423525536369016832
Faltings height: 9.8905
naive height: 143.8272
primes of bad reduction: 2, 3, 7, 11, 19, 23, 61, 73, 127, 173, 457, 631, 3607, 1787719
regulator: 86318240454.0331812220082509097044193313362421980037477293713
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:22:26 UTC
last updated: 2026-09-29 19:22:26 UTC

Witness: 11 independent points log in to add more points →

(-3961591856, 108399267259240)
(1649520112, 31298937887944)
(1696426840, 26948140515616)
(2422224880, 6312616273096)
(6244160848, 420167152299208)
(11010085480, 1096343603006224)
(-472162943034/7921, 99889423559536089942/704969)
(-17928090704/25, 21206628235490216/125)
(7150920046992/2209, 9574875102792550104/103823)
(404099797175176/14161, 8057552424115565494544/1685159)
(1412127997173832/2209, 53064430390721141243384/103823)

Commentary

Specialization at t = 3/11, u = 11/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), 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 19:22:26 UTC · history