curve #6011

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y² = x³ + x² − 2560724776044631308839541374657851945x + 1597115244114797632729206190950223395851534923925596475

a-invariants

[0, 1, 0, -2560724776044631308839541374657851945, 1597115244114797632729206190950223395851534923925596475]

c-invariants

[122914789250142302824297985983576893376, -1379907570915185155415522884481846830961514090173176714624]

rank (lower bound)

≥ 25

torsion subgroup

trivial

conductor (N)

38394771744510785556176953859518468466663700500649746238666433145553214754846053363273466449179240

discriminant (Δ)

-27281646950124129094890538479690162735339232372049831283285737670101653729308818747410150208302938811212000000

Faltings height

19.8028

naive height

263.1387

primes of bad reduction

2, 3, 5, 19, 1789, 117787, 5567363393160807858848091741206387951, 14354232369553341141842215891678113905167243259281

regulator

1502452129112257142854331917.371061474022946794019980233497818531687320668918314828762403

submitted by

apkorte

submitted at

2026-10-28 02:18:24 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-28 02:26:33 UTC

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

(773778971875165085, 281012382545415061504779450)
(930652906644693335, 141497613626744400106981050)
(238498688545612565, 999975967777574757182270670)
(10553432833411045175, 33911070097424748870793629690)
(727441974511520735, -345366985993446407657167350)
(53777727124925285, -1208122846988744295975045150)
(41888637679314683, 1220624214083284142371366746)
(25203704670584821007/49, 221244516515392271188696015398/343)
(1051990028389525070, 259764923952519387995614635)
(528980111368356575, 624949565615528357765785590)
(109661839787444645, 1147876394245623784899965790)
(1019816306030033579, -215130471758770908797748090)
(8744451748223562485/4, -20312300374780914972109247535/8)
(-1830283224454696435, -390683929304411430011827830)
(-223718227172934715, -1469285196501256051683652350)
(-1436309438997028165, 1520534071744476350817087450)
(1818678803396060135, 1719135802126332746079428250)
(881449405972932035, 157514191310953622901940650)
(918258594804210035, 141358756035330526053511350)
(-1769066325227454865, 768599319870019138714598250)
(-318980724144401210, -1543204851193604198882885055)
(76314646307313812935/81, 105201767870563601676934974250/729)
(832485000358054055, -205641483928989748013926170)
(5671115901814540973, 13017943673135271492584086506)
(980952706243875755, 170601734139081634041642630)

Commentary

Specialization at T = -71/67 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -19381964681583531802332T^8 - 28905723289750482456822T^7 + 237762949204059961327281T^6 + 531898331411046911194404T^5 - 2081641537694219583469905T^4 + 2900998586783373204925386T^3 - 3510952046475025537069569T^2 - 2402026805831481679416408T - 336697228013419606811427, A6 = 1037857170981138962547584146233744T^12 + 2390096821086188960269294192921236T^11 - 17622524818710876412253124148231989T^10 - 68498401439427972797592193690386720T^9 + 142403571107393724072842085683710290T^8 + 537652195534688333016592840720348080T^7 + 242869176165494440760869297647114570T^6 - 5350043176257097379858948426236938120T^5 + 4432615889801018420373294120358506840T^4 + 3400474115253102291019214783235467100T^3 + 524125620210135846436694485054404435T^2 + 22638281992604199603670162474986120T + 4060904829303617481967199289389790. It is a 2-neighbour of the fibration given in curve#4205 (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 2026-10-28 02:18:25 UTC · history