curve #5967

y² = x³ + x² − 478391468905886027265201332584406154891905x + 127078741503594653802792480475873926667515102857287725962414403

a-invariants: [0, 1, 0, -478391468905886027265201332584406154891905, 127078741503594653802792480475873926667515102857287725962414403]
c-invariants: [22962790507482529308729663964051495434811456, -109796032659105780885750479874199967816585426852480904204134912896]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 11453057263688698098742535669886806309000076820212260831211775517718624305960064358248066360
discriminant (Δ): 30599091362320288357785145810551400413454704352254194539198511337599465135423117091178600195992340627097818074409846932000000
Faltings height: 22.7634
naive height: 299.5273
primes of bad reduction: 2, 3, 5, 7, 13, 17, 23, 29, 53, 79, 157, 167, 192097, 2488949, 18563242045751583918041, 3273531681496498421689375926154260842497
regulator: 39304166203777634089884255.93686422827581722250500862028147980950404414159020311843917295
submitted by: apkorte
submitted at: 2026-10-27 02:33:31 UTC
last updated: 2026-10-27 02:33:31 UTC

Witness: 25 independent points log in to add more points →

(3602435210060898533551, -212495763482503931933852626163250)
(236340280576014092161, -5216968692394914838506738393870)
(28243280036724276571, -10657857498611322415670962909110)
(445657333572671031313, 1546700562490240851642221329602)
(-212806122830110288634189/961, 444031907639804323097671812542779350/29791)
(-113778460646701765049, -13417766692864652767300962259950)
(414789292870653250051, 107914596971221888853648438250)
(155569004273862943951, 7511384276628692038589439808050)
(-797234426270712020999, -1326821882054410182072505541850)
(-641147386434665461199, 13047671970592584987814732193250)
(373362014832557086726, -715473812797965370379554989525)
(280019446222457639446, -3882834193240866796064943354765)
(299083094429856586351, -3279196853472524006332928012850)
(383966458197008139331, 27037600750490460551388954090)
(162055822402389386401, 7335430669276849674734318572050)
(379174182642168416851, -447303148135371616072767023850)
(157480791184194153793, 7459680257075430282017566069938)
(311670351930008079601, 2872884599404100172312502413150)
(39544698882072167671303/121, -3160836386140562218499709696591222/1331)
(515515173176257230751, -4178740978636424070494202838350)
(521518064109563087161, 4408174328044536745274773391130)
(484001516155337153251, 2986220813362387972095229756650)
(199420175507082952651, 6293524719668003299836953685450)
(520895340331597131451, -4384336419617026378434733183050)
(414661859412507277891, -82788920875138071265158180330)

Commentary

Specialization at T = 434/205 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -5184596861019629847612T^8 - 133754925365232341831388T^7 - 1035437637468603823145739T^6 - 1575565271267685915981924T^5 - 1170680387853697770352680T^4 + 900828146471472012207744T^3 + 1832423192062073064608256T^2 - 1187684595807033661235712T - 1681976130623118064380672, A6 = 149744321704475419746038421127284T^12 + 5524322875536651418776153811707324T^11 + 73960445444599888122367809067169241T^10 + 447433713175257084856032889546498740T^9 + 1270478308377182851928891301866983125T^8 + 510043107515189769341895147322518288T^7 - 1365617314830659026227384880724910192T^6 + 394471188044464447389716164705234752T^5 + 2190221113813265570364601521374517120T^4 - 625950896394979706215937393243105280T^3 - 1449978984373691606292352028569522176T^2 + 834866943439151412261391613342736384T + 852857001347003013712347074940788736. It is a 2-neighbour of the fibration given in curve#4191 (itself a 2-neighbour of the fibration in curve#3900): its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched 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 2026-10-27 02:33:32 UTC · history