curve #4980

y² + xy + y = x³ − x² − 4376966079644951171x + 69589513228480311626868042051

a-invariants: [1, -1, 1, -4376966079644951171, 69589513228480311626868042051]
c-invariants: [210094371822957656193, -60125339428461564572410678879425]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 679633104414005058194982
discriminant (Δ): -2086679948138825945055188048836446254209751722498084495342656
Faltings height: 10.2535
naive height: 146.3480
primes of bad reduction: 2, 3, 13, 19, 31, 43, 73, 241, 307, 691, 2131, 14419
regulator: 24027405.6842961100880236279274622775481832595417989294
submitted by: The Wren
submitted at: 2026-10-21 01:05:09 UTC
last updated: 2026-10-21 01:05:09 UTC

Witness: 8 independent points log in to add more points →

(-1063441025663781/330625, 42676231069778921660022/190109375)
(-6090891407065/1369, 1595843281111838166/50653)
(-2862470086285/841, 5176973159261659458/24389)
(22540700661/16, 16469146378606535/64)
(667043788509/25, 544114109158854492/125)
(3453357785952071/60025, 202839981769368805706634/14706125)
(5933543195185871/942841, 494103017644549146675858/915498611)
(16156077616662569/289, 2053543311557265416833944/4913)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, families/family_Z4.part2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Eight rational points checked exactly; PARI/GP returned rank bounds [8,8]. Rational torsion C4. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 01:05:10 UTC · history