curve #3172

y² = x³ − 2241400793004128647394643238117533637889417292x + 40774547849544973581800981372433536925107999761595109335509492547824

a-invariants: [0, 0, 0, -2241400793004128647394643238117533637889417292, 40774547849544973581800981372433536925107999761595109335509492547824]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 95459198382057840988784023657339466333096519203745062080
discriminant (Δ): 2445933922975011439465755939333463829156501309135612060671674220903734263316566397136626937880842163569384850269980859496827954176000000
Faltings height: 24.8668
naive height: 324.8839
primes of bad reduction: 2, 3, 5, 11, 13, 29, 31, 37, 43, 83, 89, 101, 107, 139, 199, 211, 439, 521, 1109, 1231, 1303, 10993, 199673, 389461
regulator: 4931959490442.1595013236698017343355626322989748885266560101760
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:40:44 UTC
last updated: 2026-09-26 20:40:44 UTC

Witness: 12 independent points log in to add more points →

(20698136552699007358133, 1802519341546011002190400906037025)
(-12639425973343582744792, 8190564976169068564057598492809500)
(1672713813713336562708, 6085228507344789506702040460847000)
(21768378070621916955158, 1515948985738596662719425367178400)
(30419413532795218967108, 860681267992582782880942238569200)
(122343554400210398554457/4, 7298469374913191741919339352111875/8)
(26359932137087906791891702/729, 52093846333020283584128963946604466720/19683)
(8389393781171822909915912/289, 2032682714372102094821266788050772300/4913)
(7599281910580065215973, 4917353244391814489675155359913495)
(-1260172694603470380537994/25, 633441974992982834318081165167176096/125)
(57405502790318115083079718/1521, 187032801571048599680128735702954820800/59319)
(-44993181106077147334522, 7109058880347357568412037838818240)

Commentary

Dujella - Peral (2023). Source: https://dujella.github.io/tors/z2z2old1011121314.html (curve entry 9). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:40:44 UTC · history