curve #3158

y² + xy + y = x³ + x² − 9616802040765253686519863489400032714x + 11458141648950677597857694544338957425465840088897243351

a-invariants: [1, 1, 1, -9616802040765253686519863489400032714, 11458141648950677597857694544338957425465840088897243351]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/4ℤ
conductor (N): 13234498274885365856717453272852852367581218
discriminant (Δ): 204079721023528757828417039132845570355951282046410513434057728508222768827871131411245199409151716839394443264
Faltings height: 20.0522
naive height: 267.0833
primes of bad reduction: 2, 3, 7, 11, 31, 37, 43, 47, 71, 113, 149, 179, 199, 227, 233, 431, 677, 2887, 3019, 308213
regulator: 32516646320144.5582484771811924853633200564390613811546267164
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:04 UTC
last updated: 2026-09-26 20:40:04 UTC

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

(1084146104289492873, 1518683309539256514245998651)
(1375161852338459147203/529, 24411918982693235834049597225355/12167)
(15422310165617800073, 59424785067732305697277231675)
(14591018719842282655/9, 9671991429426159057884111189/27)
(13841374557228493796289/841, 1601390214571428264506169246227495/24389)
(10405602055730261253, 32219128799494318875079868191)
(2749938751531443734567847/1437601, 428407644928952276034519608229665825/1723683599)
(57281960365991295632024633/3073009, 427884710327425088879593927830812916179/5386984777)
(-70551359216673824299171/303601, -618733463976659279322191600671118035/167284151)
(161949936045576405934466353/247464361, 9083669727279833423555403950536285817641/3892861862891)
(-955960764580198934391664725617255/493884441167121, -52432633421565222424864633837673543172412771585123/10975846311006366381831)

Commentary

Voznyy (2022). Source: https://dujella.github.io/tors/z4old9101112.html (curve entry 15). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:04 UTC · history