curve #4586

y² + xy = x³ − 31227770360946519543132422824985x + 57338623413199622046182069134417440709372418361

a-invariants: [1, 0, 0, -31227770360946519543132422824985, 57338623413199622046182069134417440709372418361]
c-invariants: [1498932977325432938070356295599281, -49540570629004475696300773720286075878432212862825]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 8917119221745677058307104421347510559180898815516214720856080386319147814
discriminant (Δ): 528665845429798837652124830665293323627820929530888796631863325982429548450550339772692063875072
Faltings height: 17.0905
naive height: 229.1702
primes of bad reduction: 2, 3, 7, 13, 17, 19, 43, 60373, 990488237, 567570027575987, 191026498937557279, 13951228500613746619
regulator: 4975707355249486941328.852716873558057302087017412908310792619952281746303139149034
submitted by: apkorte
submitted at: 2026-10-18 12:01:17 UTC
last updated: 2026-10-18 12:01:17 UTC

Witness: 22 independent points log in to add more points →

(-446211786590630, -266803259948038835813741)
(2143098202723570, -16044193777650374553341)
(2048202270794746, -44388211700396909844077)
(-526112283562748, 271334318391031004610169)
(-25346990506293101/4, 222424668176839868244059/8)
(243619119614716330/121, -68217739882670283951782551/1331)
(-121216541355350, -247228972030883135443469)
(5169617404315066, 184555502459545433897011)
(4297913792947594, 50156679218166101536195)
(-17893620397940366/25, -35205439163666073740350033/125)
(-6099077307047750, 144641607045284938368019)
(-5648363395865306, 231341701497791845768411)
(4590107234933338, -103485100815692554433261)
(1497459801189978, -118043108532610167936237)
(38334144739253338, -7429167501576822118980077)
(5312873247781438, -203454919909994164583477)
(1572352950048076, -110112965562400781575199)
(11883601065754666, 1168093353019403701789411)
(673630125188869876/625, 2467223335301803220657378299/15625)
(-5479883124610724, 252798524200952673092851)
(4596136748446036, -104414978139416421239159)
(814468811654746, -180124579450990228644077)

Commentary

Specialization at T = 31/117 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -80980973155382618456892T^8 + 285465051520066035306000T^7 + 261576072020933408151900T^6 - 951872800330750729557600T^5 - 740266115548451074781940T^4 - 838297434410400458533680T^3 - 2411861402496464207592220T^2 + 321039963467228344391680T - 16364003227722838984000, A6 = 8727598530781278063259830058307124T^12 - 48350067107354860665788856770508000T^11 + 6795007788217678621771300867414800T^10 + 321740442330556067264524621932532800T^9 - 148612015016082982201355258151421080T^8 - 1249986278690606236857933073188907584T^7 + 185359528793598231769817920929440640T^6 + 3119211110386308385542728766215321280T^5 + 1959126383802129619521972397334285700T^4 - 734642679786885593606821730974577760T^3 + 302738566919351785316592492519057616T^2 - 31992857837231850069180957876807040T + 1040209640416816878466851126534400. 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 12:01:18 UTC · history