curve #4557

y² = x³ + x² − 10852751556075161952549607870002219945x + 12497952345250844456037665918318431712011680038995850043

a-invariants: [0, 1, 0, -10852751556075161952549607870002219945, 12497952345250844456037665918318431712011680038995850043]
c-invariants: [520932074691607773722381177760106557376, -10798230826296729613142135801576771641512378620253053781376]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 143917957977217058693436268679010221745477195815652119315932364093686471734960924840
discriminant (Δ): 14330825284860788285747106080655833798875070037187364094619187516440697344698819568822963764944539150577076000000
Faltings height: 20.2567
naive height: 267.4460
primes of bad reduction: 2, 3, 5, 7, 13, 19, 23, 61, 67, 71, 103, 66387491, 21682994323693, 68663616677926063, 1458412826058344054194326737477
regulator: 189203803055899682250529.07190687929349625278628735187204577856446334405072608822638203
submitted by: apkorte
submitted at: 2026-10-18 01:44:17 UTC
last updated: 2026-10-18 01:44:17 UTC

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

(1389741727266657291, 315520053265366645573880730)
(909508178723125053, 1838378598653703880671863238)
(2512205040831294531, 1043349766635514852137093750)
(-690511658695149219, -4434260195283940606991883750)
(-3389866291190140989, 3214610336428879164963829110)
(22955287873232819631, 108901559330513463709686213450)
(-84905453131330669, 3663167762451770448944400150)
(-1730935736894895459, -5108546232661807045904296230)
(-1860314779053008619, -5123413205729398661423274450)
(2550080447835273243, 1185540881853736405728152058)
(5054553551074042579/9, 69259097976167049826279275550/27)
(2444988868107404691, 761030280622267880396829930)
(779680791155607231, -2123732573610018702001684650)
(10899383870841506931, 34482155567451618611116718550)
(2836996960182651471, -2131301486319493964124782070)
(10632960820643331549/4, 12477173312736773742634062075/8)
(9472793618351204031, -27563055418972565174507438250)
(2405362868251675131, -556798281333894654401979450)
(-3328783012259680449, 3426199167221743157876813910)
(1298267613449879781, -772272609062830126174901250)
(18105662516066890221, -75837390034387450453872954570)
(-3758055332354304819, 456347839496730797081097450)
(3416550881574812781, 3911490465129675044910861750)
(-2959182554929992969, -4324392427837480720681196250)

Commentary

Specialization at T = -86/139 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -129423262682594003679387T^8 + 364444960853480119379208T^7 + 145088936906457036995136T^6 - 946405361864597551050816T^5 + 1086632441482958356509720T^4 - 615454597853371985855184T^3 - 2089672621773109442565264T^2 + 1503858169634937349756032T - 268043453574349038086592, A6 = 22378678518203765279639595291703734T^12 - 97642937907003790975195419097186584T^11 - 57701568157490976356217548246274864T^10 + 692728062356311676289756774010506240T^9 - 165059119295086271477298303488050440T^8 - 2065127205357525454766777049159397680T^7 + 866901377553305534044230525750471600T^6 + 2900595200522009433908965902519647040T^5 - 1511852767062133534934112666639840240T^4 - 1121738997234616988726901721424515200T^3 + 1300199883149489416130125996732379520T^2 - 448873241175573293059242892011302400T + 53141649442568035635166156713288960. 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 2026-10-18 01:44:18 UTC · history