curve #4268

y² + xy = x³ − 28465543849432816324623481891607269501565227730x + 1912392754754874216866607120233635758960309880423899579727165768593156

a-invariants: [1, 0, 0, -28465543849432816324623481891607269501565227730, 1912392754754874216866607120233635758960309880423899579727165768593156]
c-invariants: [1366346104772775183581927130797148936075130931041, -1652307340108211323372750601401018454904483109576945432607675336760883345]
rank (lower bound): ≥ 28
torsion subgroup: trivial
conductor (N): 13254623199995407422640596558711966162848609362761245598355101472245794255676684213067251702177179460045793902
discriminant (Δ): -103753304634559133523953656302161172553189908351064859811767624372452282326727675614470347349863123677378504676170305794066145985763852320768
Faltings height: 25.6343
naive height: 332.5766
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 23, 29, 41, 2633, 1228033363, 413206565930665426383996259338260620026299091043801666189715739305939401348860205909
regulator: 6043890484070334802354174130283.163081313470506468746506527844937793372855914284051265581974701
submitted by: apkorte
submitted at: 2026-10-16 00:28:16 UTC
last updated: 2026-10-16 00:28:16 UTC

Witness: 28 independent points log in to add more points →

(618909920991125770930714, 470497518630730070220324426734957170)
(127034177755142142970660, -18610024224135281799610001406957314)
(110184867717260110881988, -10660326156599217108354165359377826)
(78311794873294048023748, -12785561495500595424791176707405602)
(-194121207168249218839100, 11094173178336252704911115923338334)
(427965943479555029815531636/3721, 2870064512132323961256279882468807384838/226981)
(4319289986551188608786404/49, -3219893586501545737786441887032992142/343)
(75310294173376220901356575/4, 653527910670700903498583205489682790153/8)
(-138637666599806384062262, 56516524706819593710556148509441924)
(150910152076489555803844, -32457079230498361077230655818333474)
(1893153182955188933264164/25, 1727782050425516821670598114564325478/125)
(434049471507710909376316, -267079389776596462944820050248446202)
(67760781422403093379236, 17165968662219866687168130912444798)
(5521284668883536360696644/25, 9998311030857444346691735975082153062/125)
(3349064478944747229480604, -6121312489230166403533247610035269850)
(-181404426862380353983484, 33265451401704296782988773291310686)
(6407142507915677833831669486/49729, 217205031761376199057957880282379677566720/11089567)
(98327696107367231434720, 8006659487223652453355971076764606)
(1494741440805808218471082564/12769, 19588183926034433094869840847648825990206/1442897)
(18685932961045851000573292, 80770828873825810242984897584665403134)
(46270681818537301078744, -26350276119942308456489658737424674)
(2972093877249190705842586/25, 1805059734419236402303057916135595626/125)
(76152370182268197787798825108/11449, 21008100669727862510534989660878192465512762/1225043)
(-1996989879531652790696156/25, -7579286296551167127450501738979014682/125)
(419987770012219683994280932/3721, -2660675364909855981019166888796464383466/226981)
(-173669409687195941272964, -40223535339512068604985845184571202)
(66066329997207421342328344540/97969, -16495814646571287090564858066257422504538258/30664297)
(-29981488566468615905708, 52334329863415820533397784530043934)

Commentary

Specialization at T = 3904/4935 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -11696112421636110725232T^8 - 225953091010175473234080T^7 - 1285862935447903797580320T^6 - 1129158151626029959937952T^5 - 1592088393814170815994480T^4 - 37631263114247380337808T^3 + 1695532195366906279824008T^2 + 289018889748535735084344T - 624976459672410496278747, A6 = 449725080902042019431313876378240T^12 + 11412860638353264300064607972424576T^11 + 108724211594005774585272702173979840T^10 + 552699339212501471676010717898982720T^9 + 1706301350149364859650702774642594064T^8 + 698629245250434441649386767885626848T^7 - 1435130050396849573556196386817943392T^6 - 607341322268570723710732507501947360T^5 + 1272148652815891722874496655477939744T^4 + 339936587466097075261357997400067920T^3 - 735579972865799521796462939416343048T^2 - 132384785109517617529019092091956392T + 190126813773008796183739475605546614. 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 2026-10-16 00:28:17 UTC · history