curve #3159

y² + xy = x³ − 204612883217644162374401645935933337240x + 1126458801999575619505157608163376285898744251881550465600

a-invariants: [1, 0, 0, -204612883217644162374401645935933337240, 1126458801999575619505157608163376285898744251881550465600]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 4669857800272842393315326009855704719253530
discriminant (Δ): 81451975596829463089164630016127887580645197731925141407491009495210902417363991282266435542767718472089600000000
Faltings height: 20.7078
naive height: 276.2562
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 31, 37, 79, 89, 109, 113, 439, 4261, 5407, 8623, 673273, 907969
regulator: 17530959748405.1266087529587000712273307306203003887777355731
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:07 UTC
last updated: 2026-09-26 20:40:07 UTC

Witness: 11 independent points log in to add more points →

(8516689742361939760, 1258557746444139996430355560)
(87151157544487517830, 803265529207451947250387612710)
(1726456402245267089830, 71732928956835871254964242920710)
(89877195839939054896, 841876686600359771888361686632)
(4014405800817444147760/529, 39864065312678465090276882922200/12167)
(7353521281977893280, 4412476003028123558085446760)
(46219346744713441546960, 9936551405074724996120320013857480)
(990903204473409123760/361, 165968122228853630339350672422520/6859)
(1147807797564561615472/121, 8321895429390224494641744582776/1331)
(226000159585908090165664/10609, 87656215263243991410412542676632664/1092727)
(416445801148637774289040/49729, 5540488986506714228645753000442520/11089567)

Commentary

Voznyy (2022). Source: https://dujella.github.io/tors/z4old9101112.html (curve entry 16). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:08 UTC · history