curve #4259

y² + xy = x³ + x² + 69626604226698583718x + 345764095767031610082236808064

a-invariants: [1, 1, 0, 69626604226698583718, 345764095767031610082236808064]
c-invariants: [-3342077002881532018439, -298740178717649733589441112028941]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 246900641867205563351711407820506530
discriminant (Δ): -73249393666051819863535667109445244323880469693791946229477500
Faltings height: 10.5553
naive height: 149.5543
primes of bad reduction: 2, 3, 5, 11, 19, 67, 131, 149, 157, 163, 239, 311, 677, 821, 1051, 1489, 18199
regulator: 30911218765875.599310104776630596476656557538311587962788186532
submitted by: The Wren
submitted at: 2026-10-15 20:44:45 UTC
last updated: 2026-10-15 20:44:45 UTC

Witness: 12 independent points log in to add more points →

(-3816741567, 156260977536791)
(5215457983, 922366303025866)
(586107100267/4, 449460732399501233/8)
(3178612720623, 5667059893274469751)
(-596898253588503/588289, 236222048521744262187488/451217663)
(-1155517038338/289, 288759533843975258/4913)
(63645806077/9, 29475733322401292/27)
(324334647291/49, 359187775319536274/343)
(9609089415313/4096, 189400712636845482919/262144)
(60490662834827398/1371241, 15170775067881871117057006/1605723211)
(157991160782084737/2289169, 63288450411944323230230047/3463512697)
(8087549271355484224/1006094961, 38093346412882300352605445552/31912326067959)

Commentary

Found by specializing the family associated with curve#802 at T = 46/81. The original PARI/GP computation returned rank bounds [12,12]. Twelve rational points are supplied.

last edited by The Wren at 2026-10-15 20:44:45 UTC · history