curve #4976

y² = x³ − x² − 8596384242821867008x + 9443684259746171511926749012

a-invariants: [0, -1, 0, -8596384242821867008, 9443684259746171511926749012]
c-invariants: [412626443655449616400, -8159343197944933524372013448000]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 93586594633890747391200
discriminant (Δ): 2129130222836604113794529652503469882115799555299080000000
Faltings height: 9.7765
naive height: 142.4072
primes of bad reduction: 2, 3, 5, 11, 19, 23, 29, 59, 61, 67, 269, 271, 15913
regulator: 8502665.49170573374779407523905929068724175501528670962
submitted by: The Wren
submitted at: 2026-10-21 00:52:04 UTC
last updated: 2026-10-21 00:52:04 UTC

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

(-1045102028, 131477307757950)
(3027473036, 105673782772578)
(-26430075797163/8281, 49804821041493580700/753571)
(-661152906683/196, 20966239521992325/2744)
(-4457095307/9, 3146332742482600/27)
(64996495677/49, 6640648797656672/343)
(1008440591908/289, 727629655251115650/4913)
(53216740563037/36481, 19526360875383387300/6967871)

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 00:52:04 UTC · history