curve #5853

y² = x³ + x² − 3154134917455225404794607220x + 81142250506431183863769633858723645865568

a-invariants: [0, 1, 0, -3154134917455225404794607220, 81142250506431183863769633858723645865568]
c-invariants: [151398476037850819430141146576, -70106904437557451249153190758853810874730176]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 712390990648684149085476851096868189413638843337350011960
discriminant (Δ): -836052141990873647723521844258707683657868889166059158297066537416473360833500000000
Faltings height: 14.8218
naive height: 201.9172
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 61, 89, 42016714997, 2980207120993379, 17476559978303266459
regulator: 9039964248284990.875684887327884900605253004657934182818555794290211649804478
submitted by: apkorte
submitted at: 2026-10-25 12:16:13 UTC
last updated: 2026-10-25 12:16:13 UTC

Witness: 19 independent points log in to add more points →

(78226154341706, 559551744135131816250)
(23602565776706, -140872698631354121250)
(-19950289506322, -368954820674464586562)
(65464357410956, 393969177324830079000)
(-54520766351044, -301735586401738398000)
(27607371864716, -122907697903241498520)
(-56779928745169, -277808512809268008750)
(-19112011965169, 366664818688891076250)
(-196782302614051/4, -2739323967061315417875/8)
(-65442836632894, -85331735762276326050)
(-6842039443414, -320004165882883391910)
(-37754732781304, 382634846860353731280)
(34457425628906, -115631618745295229850)
(78879054524186, 568440914478839610390)
(87768408938126, 693119521252548097890)
(57334290114206, -297947371895295491250)
(44091132934331, -166695127989750864000)
(29898765322706, -116469302859088656750)
(37288108670456, 124000100406544635000)

Commentary

Specialization at T = -11/16 of the rank-17 fibration of X1152 given in curve#848. Found by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-25 12:16:13 UTC · history