curve #4985

y² + xy + y = x³ + x² + 722677909237395434437312x + 243848445232480896651380113381173281

a-invariants: [1, 1, 1, 722677909237395434437312, 243848445232480896651380113381173281]
c-invariants: [-34688539643394980852990975, -210685056680603330659466955604936282625]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 50215092951230658153897413550
discriminant (Δ): -49843016132592967479488105712734385224565656602761390965928902274345000000
Faltings height: 12.8275
naive height: 176.4869
primes of bad reduction: 2, 3, 5, 29, 47, 89, 139, 181, 349, 4889, 12541, 5126089
regulator: 4386557568.5460470750075192678696473857115271622892061242
submitted by: The Wren
submitted at: 2026-10-21 01:11:20 UTC
last updated: 2026-10-21 01:11:20 UTC

Witness: 9 independent points log in to add more points →

(-27635000081, 473134388482574513)
(1211925789705, 1702853925936446197)
(-23258591894806719/259081, 55675651542668907042386165/131872229)
(889416355221/16, 34118230472953762567/64)
(17911177544125/16, 100292338224165522443/64)
(375885055107105/121, 7584044952699751429907/1331)
(78844341380762625/380689, 149000436305790945946748081/234885113)
(96059329560549900975/50027329, 1044340639042906295363468350079/353843298017)
(156951672596329499541/76860289, 2155713128414027189780939601619/673834153663)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, families/family_Z4.part2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Nine rational points checked exactly; PARI/GP returned rank bounds [9,9]. Rational torsion C4. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 01:11:20 UTC · history