curve #3227

y² + xy + y = x³ − 51671229331059644891666297654483243x + 4520596773662388877346447309051182845388785311959658

a-invariants: [1, 0, 1, -51671229331059644891666297654483243, 4520596773662388877346447309051182845388785311959658]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 53142274232491098392872393941753394804628410908321062888347710
discriminant (Δ): 1038164972670138684962915411252330778511955934523086706439214876434946233525468531001647726156046157500
Faltings height: 18.6296
naive height: 251.4042
primes of bad reduction: 2, 3, 5, 29, 31, 41, 59, 89, 107, 113, 191, 199, 397, 577, 1009, 1123, 1193, 1433, 1523, 1873, 3881, 58549
regulator: 18332353588657173.64236300117494762475962767939652062903364006112152413
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:36 UTC
last updated: 2026-09-26 20:43:36 UTC

Witness: 16 independent points log in to add more points →

(141619764967775934, 6578584744791410297700985)
(99864500008570989, 18878977611252925518992410)
(248238803992676919, 83611537644002554260799390)
(507266594870876331/4, 21686667772091179044642155/8)
(-248205283317232896, -45329551958008032415871930)
(179703173729305164, 32222800996699387743315625)
(-897824974045715049/4, -554849767967179778496192655/8)
(8527070458458717, 63879672597508782542481130)
(-257763946075387032, -26705050136250711582795047)
(132204028175118294, 318698988584720949558640)
(-255454636170693771, -32404173939906937257012680)
(129912028384611264, 652241277633523846496185)
(-122005078189427481, -94914036770170655800132910)
(77761220204880781509/361, 399061991814514109741444454760/6859)
(144584763577150887, 8499011128039330574109490)
(132062728972984404, 41787089121228963105250)

Commentary

Dujella (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 22). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:43:36 UTC · history