curve #4195

y² + xy = x³ − 77742406859070276992454593681376269361892585x + 229711807538935477005166932197499449222673317168718072399630398297

a-invariants: [1, 0, 0, -77742406859070276992454593681376269361892585, 229711807538935477005166932197499449222673317168718072399630398297]
c-invariants: [3731635529235373295637820496706060929370844081, -198471001713640252132469826871933377188333202764517473644674720394729]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 17325221244070607551070058908530094913795847478774914858418853013110279303837924712209042392405987310
discriminant (Δ): 7275852592599506101656665672248212335423785976825992875037322571408096305496452743497226283696950701970890849361293632851276800000000
Faltings height: 24.2201
naive height: 314.7995
primes of bad reduction: 2, 3, 5, 11, 23, 29, 47, 89, 97, 131, 151, 1412925975016666469, 6940835705861581881638584274637257599061774294349159013941539345039
regulator: 2225189286992116014832603022.17738555743130338322300032086932342840850986022207284814544838
submitted by: apkorte
submitted at: 2026-10-12 04:31:10 UTC
last updated: 2026-10-12 04:31:10 UTC

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

(1272752397372164609804, -364453978907399589552182469996157)
(558550491107418746133584, 417387871707006379826965023263441333)
(1546339508849269952042, 336441980981034474245932878618995)
(171494612927335271000690936/529, -2245000802781024319440251558183529265139/12167)
(-39453563981891178372289/4, -1537602361533897234767726208118211/8)
(23987972283284391626626/9, -5496147326749695119466011614746079/27)
(-5732936057978946894086, -697841093621395244809233200071037)
(-10005875523936450425056, -76347629066248952552349953846587)
(-9864858742623556285216, -191380824664463480867922821743717)
(1318395095590055614034, -359872461653324439156178330013317)
(13998689781547313589584, 1372825420319921291598081807857333)
(900312562985607460497074, -854218958529217338528071542687870477)
(468823329738043049177834, 320950079397788943732044126208858083)
(-3002611931978370643666, 660357102192250805964880371549083)
(12805455334244261580911/4, -934323034933946603861205109584311/8)
(102099058520888265292634, 32505308850024145410942607446704483)
(9073349328014908163834, 520862109567537980611238079294083)
(-3947080101982733191078, 689256129356782330272531016983875)
(15575834076243600637634, -1672605523347054591676281262027117)
(2688403777612460549402, 200347961160145282755314841973955)
(9065847988852590097274, 519643514655770522260396844816003)
(12303374056979448415589954/1681, 15834476378932958432905462004511704443/68921)
(36156031093056632950634, -6684619432132374745670951367711517)
(-7810403604386799399916, 600381530855229170982486139575833)
(-7617821571743397514366, 616334505337782008310385438718483)
(17419068227395171971657194/5041, -17291740276198240984130465357330789387/357911)

Commentary

Specialization at T = 5403/1139 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -4247232044073407555787T^8 + 31562440352380138911048T^7 - 1079699554459262036768544T^6 + 808673010384011405337504T^5 - 370492253920133220842280T^4 - 583074314969616668304144T^3 + 1749298418924942437857936T^2 + 6981774061823935535232T - 733152591313571732712192, A6 = 99353320243156880097765275026566T^12 - 1192047168420920079142624551372696T^11 + 51816962014333011197785218484470384T^10 - 160733095313938224161654547702105120T^9 + 1161258380951665950605765814917343480T^8 - 500375410396179478662655437513305328T^7 - 2173656091315178125605895871772822192T^6 + 963181788398399075637606795285979968T^5 + 1772421005485324185894755005716179280T^4 - 460279153080260982936199267860712320T^3 - 927109903404780261014801473928185344T^2 + 58145438007481575316035550439430144T + 241998303936746269490887631501697024. 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 04:31:10 UTC · history