curve #3163

y² + xy = x³ − x² − 4536184139377606759385840064x + 117756938515259070124029037981449282901220

a-invariants: [1, -1, 0, -4536184139377606759385840064, 117756938515259070124029037981449282901220]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 91108647893562681246204595773459948396390
discriminant (Δ): -16594723409064838438365652850865750050223183177314222165170493833025838989226016500
Faltings height: 14.6740
naive height: 202.6620
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 102299, 5412646898693
regulator: 692514661.594837113820407542967403333058565325437477519422508
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:18 UTC
last updated: 2026-09-26 20:40:18 UTC

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

(-19490912550326, -445832687088207682886)
(99694509539254, -810177524574020786498)
(44209842630586, 60179204272303054492)
(19042989810446, -195653002235648166468)
(39937171925371, 17130985069181971582)
(-5180338033241, -375655273818610196726)
(40648708695664, 23053923614463599554)
(264049839228931, 4162948942598699706502)
(-11554767109984, -410644344382289546298)
(8407403487266, 283220194078161953452)
(37535093077111, 19323726664997391322)

Commentary

Eroshkin (2006). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 43). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:18 UTC · history