curve #3662

y² = x³ − 5679832676302668x + 164482318080161433485008

a-invariants: [0, 0, 0, -5679832676302668, 164482318080161433485008]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 1012399021707226560
discriminant (Δ): 39476172672100594521611988742142189285113036800
Faltings height: 7.8296
naive height: 120.4407
primes of bad reduction: 2, 3, 5, 11, 29, 61, 101, 163, 443, 2477
regulator: 814802.6112118405520630896109011864729018325394862835
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:28:58 UTC
last updated: 2026-09-29 19:28:58 UTC

Witness: 7 independent points log in to add more points →

(-7354596, 453715089080)
(40983573, 23252371681)
(45200709, 9980366935)
(45272429, 11512700595)
(-3691258616718/51529, 5280049522681380392/11697083)
(1988518002/49, 9808794920408/343)
(82620195244/2025, 2345279850848528/91125)

Commentary

Specialization at t = 107/256 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 2026-09-29 19:28:58 UTC · history