curve #4256

y² + xy + y = x³ + x² − 7049987678870650225685x + 227185865604025494656339984045051

a-invariants: [1, 1, 1, -7049987678870650225685, 227185865604025494656339984045051]
c-invariants: [338399408585791210832881, -196288587884416022947471180296170825]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 18788294275983832628734782745815765808134
discriminant (Δ): 128654047501899678878280379249418350976525117761581228757899804672
Faltings height: 11.3581
naive height: 162.5355
primes of bad reduction: 2, 3, 11, 67, 103, 113, 131, 179, 197, 211, 227, 283, 307, 8971, 493729, 4287457
regulator: 711768356466864.1381737231514213477684847043902007689559728757286
submitted by: The Wren
submitted at: 2026-10-15 20:32:11 UTC
last updated: 2026-10-15 20:32:11 UTC

Witness: 13 independent points log in to add more points →

(-95975510169, 4444477300272652)
(39899215719, 3068259652140556)
(51493629011, 834602579775992)
(58255203303, 3766436170958092)
(48362349885927, 336325972571519525644)
(-21613586001401/225, 14320617053274107188/3375)
(415671843043/9, 8399773495648588/27)
(14490463337999/100, 47415364817629643687/1000)
(59257363418671/9, 456119099208099919012/27)
(338995183696631/10609, 6421950839751948888212/1092727)
(2581227902966031/40804, 48311244504031328192407/8242408)
(241644493600470543/4669921, 9751165997030766316256884/10091699281)
(10948238462516005495383/239560323601, 78156137365684123830408487358636/117252560826185849)

Commentary

Found by specializing the family associated with curve#802 at T = -37/54. The original PARI/GP computation returned rank bounds [13,13]. Thirteen rational points are supplied.

last edited by The Wren at 2026-10-15 20:32:11 UTC · history