curve #4279

y² = x³ + x² − 262568662251683935473874481611009900x + 53793899960044838430278277233936436925738609357115984

a-invariants: [0, 1, 0, -262568662251683935473874481611009900, 53793899960044838430278277233936436925738609357115984]
c-invariants: [12603295788080828902745975117328475216, -46477929565478740479380206258606054920314009188519061440]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 2102111046951998842720815252641704571627950736698543481130090482954333028675210882645528
discriminant (Δ): -91580907476148768918937508710933634725319114574583338449936512532943814365576353536082185855487475177370368
Faltings height: 19.2879
naive height: 256.3571
primes of bad reduction: 2, 3, 7, 11, 17, 19, 23, 193, 4337, 12659074067051, 77273938990273157139893910725586978609786919452643150244777
regulator: 3441781597093217147433951.1502169580128723106031593732078807390980199795896029766106014
submitted by: apkorte
submitted at: 2026-10-16 13:55:13 UTC
last updated: 2026-10-16 13:55:13 UTC

Witness: 24 independent points log in to add more points →

(-335703505699289974, -322655225961629204796288906)
(2137604185946276510, 3043023374027517009944119578)
(136579392946298156, 143108958410493739825569156)
(402800389968850067, 115692524298079898745075456)
(1198745244530938166, -1208977289696450037236151294)
(574203208338442640, 303885313257428081276043828)
(10790370982800500242/9, 32650744177361753239757581610/27)
(-546083168553650446, 185290307477435537177269458)
(4896741795549134156/25, 12424263168188794347358560396/125)
(-560216850941364016, 158332550999033611931958612)
(-19774660801978966, -242854634637916392996703062)
(192940568988922256, 101568551600355459162270156)
(-58883967640134810151/256, 1308281263588305599380697142243/4096)
(5954706253981476914480/24649, -258484082900819897354308489833804/3869893)
(626147461610879879, 367253285120014727075392842)
(144103536555770210, 137656298711655063955193922)
(-233027161122264304, -319883992656767154779391756)
(414698918699212880, -127376945680067405679003372)
(57616847313121004, 197121262549109125536016092)
(626567418084303725546/1849, 4856135038792195392152618687694/79507)
(9320790938743823066/25, 10986406415846893234444788414/125)
(26171835115876629323/121, -112313945888585997523469452380/1331)
(4443273580963927106/121, -279831670568528050819764337014/1331)
(245607305290197836, 64194043161076025542947756)

Commentary

Specialization at T = 54/103 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -213546515196627504820332T^8 + 869465515207593907398960T^7 - 510065089965030971860884T^6 - 1270257582515056045578720T^5 + 1426065508735989008931900T^4 - 690890144029437273227280T^3 - 1389292332816820391566956T^2 + 1307996517837724132656960T - 286267614586960854758400, A6 = 38855172470975738993518813988244756T^12 - 243637332934783116837818686074203520T^11 + 382613702852074064448992630465758608T^10 + 438404888400781537256687728554749760T^9 - 1340270054837389026155362835012033304T^8 - 438466811942831565597983911599431040T^7 + 1981768199902501016022471226901183040T^6 + 697917175039489822346233336938891840T^5 - 1626320407404938608434545478196300764T^4 - 187748176254897658526766053163492480T^3 + 919892555571017546557734659258394000T^2 - 417938005135807422838969764298752000T + 59851413200989840657350337781760000. It is a 2-neighbour of the fibration given 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 2026-10-16 13:55:13 UTC · history