curve #4188

y² + xy = x³ − x² − 11174164252415787452433155739542718102804x + 267886234291314551527349387137650516425898429477051887088860

a-invariants

[1, -1, 0, -11174164252415787452433155739542718102804, 267886234291314551527349387137650516425898429477051887088860]

c-invariants

[536359884115957797716791475498050468934601, -231453706427695772517216251008408236102250681428431603334569349]

rank (lower bound)

≥ 26

torsion subgroup

trivial

conductor (N)

3804257995298578259616917826902327465995028217058588411971726342268947602255635322783133001109035897530

discriminant (Δ)

58292954798080612901244548857378740130624488912401805971881859209276663674676063504678827502263527489417238142041654562500

Faltings height

22.0642

naive height

288.2569

primes of bad reduction

2, 3, 5, 7, 17, 19, 23, 31, 43, 67, 127979, 346259, 352501065480894646588728187, 582631398291353308547458494022671980775866631962413807

regulator

45837057303746887049943376771.4726039677296897834178345643811978265155543242528268815913233

submitted by

apkorte

submitted at

2026-10-12 00:38:22 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-10-12 00:42:51 UTC

Witness: 26 independent points log in to add more points →

(218345110680003348046, 2870117620671745632727060067302)
(355647576303775733521, -6424793856517688830586928375848)
(459164482702694885059/4, -5644362304258849506737652451459/8)
(878500304218221667246, -25854307581900822040748872037498)
(91129017299984263171, 79852050414653270297038189402)
(103444994459580074446, 467895276793377682950570880102)
(164830788141920044121, -1704217921460981223190798862873)
(1355828420211149139946, -49774464364648723776318845072798)
(146858726726108049439/9, -8095278376728952809682465433671/27)
(112504415673278386310839/529, 33344451743615197448380677610881629/12167)
(3083083848866894420371, 171089978592903940632001201025002)
(-66976424403008679554, -846076936462398490256438932298)
(10148603828176173755569/529, -2994569785772130588103692734450536/12167)
(495575772520885001539/25, 29093046216635148491918599177703/125)
(959991438223526971219/4, -27022646344725170112914632051639/8)
(1012503270715721628940, 32045822980386150078549546958210)
(-20997501465219085459, 702323337030034170564954423087)
(187481797388029736821, 2182386558373399034222366707852)
(21162673899231152356, 202210111385274405875046336382)
(-111574783770483793354, -354477441866296031677630643348)
(199264709738091774514/9, -4779666270894580466956334301796/27)
(25410502756338949366, -18775064246356971332793262178)
(95248063389407253316, 260149184822003029052222138662)
(4330905560062787772301934/29929, 6713156400568311888219497992931533884/5177717)
(357436855422593654986, 6476137424093133202495209037492)
(-74252069698295234650241/676, 7247782584880008672832512421486427/17576)

Commentary

Specialization at T = -257/517 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -113946597679617331526187T^8 - 628704898397157619416528T^7 - 910139814353468861642484T^6 + 13879376823678260295456T^5 - 533762798296920925528530T^4 + 1639003787601085356814224T^3 + 806892413125232101174956T^2 - 592355532948433669282752T - 295458143973808062849867, A6 = 15882493844523546552540807840095910T^12 + 145809795943145321517489625166050800T^11 + 489200782557326126999832807296770620T^10 + 474439873253302462549441686771960000T^9 - 939321256931213164482716468158448790T^8 - 1775504390224299025107326496664527360T^7 + 1396226578858524162480471413923945800T^6 + 1631143206019716771050709031938957120T^5 - 432541045559235732878122118765459190T^4 - 878384718510400823408137143448720560T^3 - 116676491482650081744926274060252324T^2 + 185829851435689714140802850197329216T + 61035568685910358648726719363207174. 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 2026-10-12 00:38:22 UTC · history