curve #4981

y² + xy = x³ + 17169621911336322126x + 9068288692652609194587887955

a-invariants: [1, 0, 0, 17169621911336322126, 9068288692652609194587887955]
c-invariants: [-824141851744143462047, -7835001429215641566507720000049]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 1349320766930761037785491
discriminant (Δ): -359463230862002161703834184663950532004149748016654531763183
Faltings height: 10.1166
naive height: 144.4826
primes of bad reduction: 3, 7, 19, 37, 67, 139, 193, 2293, 3943, 803461
regulator: 36830136.1680030125765808777743966558735070920783686763
submitted by: The Wren
submitted at: 2026-10-21 01:06:24 UTC
last updated: 2026-10-21 01:06:24 UTC

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

(-411140187, 44041885062705)
(347335293, 122775261607122)
(725515587, 148009801528053)
(12110692458, 1411824635281275)
(-1426352190923169/3996001, 429738436913302909651572/7988005999)
(664352883063/676, 2882202659817175557/17576)
(1216602507579/169, 1562390911791847272/2197)
(4581617274039/625, 11362866338850863037/15625)

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 01:06:25 UTC · history