curve #4207

y² = x³ − 125952811003880197936758938301577153092x + 370490508011032964765392411333712295846849448379805746724

a-invariants: [0, 0, 0, -125952811003880197936758938301577153092, 370490508011032964765392411333712295846849448379805746724]
c-invariants: [6045734928186249500964429038475703348416, -320103798921532481557299043392327423611677923400152165169536]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 2298247654048435327349839964710891160783031039915487120729729326813369236867188725107851658520
discriminant (Δ): 68582567204242752608347261598231535636763609884677142368780161034077040120811678063349721022871229834170502772000000
Faltings height: 20.9313
naive height: 274.8005
primes of bad reduction: 2, 3, 5, 7, 11, 23, 41, 47, 59, 79, 107, 4079183551567, 64903365949703, 6495083051416390583, 11541019832122197760104771350479
regulator: 62681866640638169886891645.09203701394498239065179047258539158560021160053615933029261532
submitted by: apkorte
submitted at: 2026-10-13 22:51:09 UTC
last updated: 2026-10-13 22:51:09 UTC

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

(1904134729006247808, 11728736632541369575110495750)
(9281692896433979553, -1025376622840593668450332065)
(43434261185653795758, 277200431253422566671921065850)
(9835760512847403358, 9120404864335262973377829850)
(79435266124724243958, -701141117591925513699510340950)
(14415377173114075558, -39375033737445445727139685750)
(2126951864886336558, -10593257801453118259777328250)
(-10757931779222429592, 21918793375450906526760022050)
(10542646251048637488, 14642479008114564153776993610)
(1857508623772637598, 11955798112492517659868075430)
(-3752828423254701567, 28112559613218846044059328625)
(3061000223853438288, -3691835262646360109801288790)
(313106047273401643002/169, -26317941382020644364201642475950/2197)
(-6360644248979103552, -30237293003930591949315276870)
(2870483322234641128, -5709370620127827517502365090)
(9478158041358008688, 5307045823097472068087732490)
(-69083457431291999682/121, -27989496538465354158561827881650/1331)
(26227361579332674600, -122915473819524218721648901218)
(24411716228090874918, 108827783208585269391251304330)
(1905123211728872920, -11723886655547986140149022578)
(-1245725603931842221482/121, 31943714644642417420138592706750/1331)
(13912869581483144034, -36210647320214883807567051930)
(-8979155760934921422, -27883576611862023119298770490)
(2486402305437461564913/256, 32609194514991975573932198960055/4096)
(-4007784356536177455, 28476443543290699201876487703)

Commentary

Specialization at T = -148/49 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -37778694267449139654832T^8 - 165004421248548489849920T^7 + 23964366446631678311360T^6 + 575308331009960686683200T^5 + 644216561885971678915960T^4 + 812460692539323896913800T^3 - 3819953879838341796444940T^2 - 3397350178704392175665600T - 728947575597452632980100, A6 = 2901058156988098667187936175477120T^12 + 21730259143975499723992766217644800T^11 + 17535187883067791568400114163932160T^10 - 192962391515084787078631234218208000T^9 - 223683722196314018130030826199045280T^8 + 1043994033704842065087271937128639200T^7 + 875242041280405478572518252943970640T^6 - 3146689238453170755654295705800631200T^5 - 1529213271890985767861180186700707580T^4 + 3040571437100170751919897956070815600T^3 + 3919924516140400157697908339646928136T^2 + 1660827326525159620863953893304037040T + 238239533579990858915246998091788900. It is a 2-neighbour of the fibration given in curve#4187 (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 2026-10-13 22:51:09 UTC · history