curve #4270

y² = x³ + x² − 2996151431685868949949584312218681105x + 1993869692914513542756041558938890928271475367597015475

a-invariants: [0, 1, 0, -2996151431685868949949584312218681105, 1993869692914513542756041558938890928271475367597015475]
c-invariants: [143815268720921709597580046986496693056, -1722703414678139701804111519248732019612034999522801528704]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 47796897503943459568876405605909069872285309750145030520646927454057927262243899244920
discriminant (Δ): 3935137512814823813464325117725198949797434070734424987458290504133331623301831232500241882964715715028000000
Faltings height: 19.7438
naive height: 263.5848
primes of bad reduction: 2, 3, 5, 11, 17, 31, 53, 67, 79, 89, 5569, 3092311524580964153545812774889, 159803738064166758505658320052397855293
regulator: 980661499815342142095122.23957893768277760215347500834029103093363578877909447732761159
submitted by: apkorte
submitted at: 2026-10-16 02:51:54 UTC
last updated: 2026-10-16 02:51:54 UTC

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

(939343313808670175, 91100770856231263657557390)
(939599982728767655, -90608745946913680951564110)
(968486154716814715, -23346268840323494142651750)
(965763749955863195, -32599602933293297174431770)
(18668115606670213685/4, -75752879012408881689676728975/8)
(1168213962181177115, -296670232851540887441701350)
(968781220205709590, -22170341752648786040095125)
(1039535808855597215, 51200153073745624577220750)
(1027018581538934015, 5654997469089131362110450)
(1041154495888081265, -55022066257507238958309450)
(1636872793335601985, 1214623484340856484947057830)
(1649852092806030323, -1241603825405946012783317466)
(-17594160670405461215/9, -16645597688722639046469246550/27)
(1090800037168409555, 153460958733116780484844710)
(970758036138104135, -12082810188358249392485070)
(7034835416240937595/9, -9716240512574135200265597630/27)
(1048391578971882215, 71014470400950541230300750)
(132453241799791163, 1264651518863058607865112474)
(22406262183669079334/25, -21100245304423219219725025473/125)
(8391618128543737715, 23828193517964433172900364250)
(12626761404426519590, -44466967432355064773074333875)
(1390002026870384945, -717526221420857541243466230)
(-866908718138165815, -1984880588838458550216380370)
(958661139518021465, -51139279638313663403153250)

Commentary

Specialization at T = 79/121 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -25598589065575488253872T^8 - 219912513106757586234048T^7 - 324549648424331055742464T^6 + 1523547434916270832355136T^5 - 2849165156820063127691880T^4 + 3746564993881929295112904T^3 - 973581348748449447744444T^2 - 862126587120769476566112T - 118556909623229874930432, A6 = 1625048868702522016795790220948864T^12 + 18521876511799709835187838396715264T^11 + 69457772033476867393403462775818496T^10 + 74100738882710387630246485913752320T^9 - 324420440712638126863027033199077920T^8 - 1791949805594601002560679464033967520T^7 + 4689003940396889037978292829416265040T^6 - 2864400878761409044683373393741992480T^5 - 91393345783003551423720760724759820T^4 - 338626380479684457921000808774737600T^3 + 374448354510478802333765563563322320T^2 + 177989462843015858245005305484080640T + 17806843321561611911434592141844480. 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 02:51:54 UTC · history