curve #5968

y² + xy + y = x³ − x² − 1760164358678708719333652669967177x + 27288322826260108837768103371025961276261237251529

a-invariants: [1, -1, 1, -1760164358678708719333652669967177, 27288322826260108837768103371025961276261237251529]
c-invariants: [84487889216578018528015328158424481, -23577110921888733655636139837965347166620732272411121]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 2022780087364476965932436048653444994200721557816889611842167404345336970
discriminant (Δ): 27321516604345956554368929704304139960337809768843534469060624205002172796305140651162790148480000000
Faltings height: 18.0341
naive height: 241.2657
primes of bad reduction: 2, 5, 13, 17, 19, 23, 29, 37, 47, 61, 79, 421, 122509, 135479, 952129, 1745813, 2821513, 1189954989258699414085099
regulator: 146006051290017832974.1541371680498824641279622664140922852461982013413701179264170
submitted by: apkorte
submitted at: 2026-10-27 04:44:02 UTC
last updated: 2026-10-27 04:44:02 UTC

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

(28103269209227287, -133176978012471691306644)
(28200145188832487, -278472161714686548948644)
(-844395883121001, 5364139848185159182205052)
(117940528543814833/9, -68691807933012018111554638/27)
(402456193809516159/121, -6167381184365243170815358412/1331)
(28112171628649287, -152204778342892412080644)
(26160400091304237223/729, 63279573663233042478696064148/19683)
(-36868932874444713, 6485914952114529709119356)
(18376465072654907, -1071613080081802055363864)
(129512858198964887, -44404039866350051911060244)
(1176532039573287, 5021858385659630907747356)
(267169945370864247, -136482809134711220244157444)
(3647815939837287, -4573413071702493602020644)
(80613952810694603/4, -174202857836287857981487/8)
(-45742165171527113, 3477589735220930385775356)
(2663574668651916343/9, 4305486648517665596441553332/27)
(32860631466393127, -2220768744671960878880484)
(20073928146464787, 209647470816730314391856)
(213520120158358087, -96881473144815826272592644)
(181088517709787323/9, -3639018818638772694939488/27)
(35699107687977887, 3154030291813892198075356)
(29432336805281017, -989293082432833391742944)

Commentary

Specialization at T = -7/13 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -2315727727754537200T^8 - 385085454700810899360T^7 - 21602016129394505916480T^6 + 522753273151467123362400T^5 - 5286101242963131414221040T^4 + 36372271295130380430902640T^3 - 119866073107198500927623880T^2 - 44611766420397375005288280T - 17644996097302241306504187, A6 = 1527906580671423711878454400T^12 + 405036291873933234226307839360T^11 + 29619810420503480598512683442496T^10 + 83447431068127636496045716863680T^9 + 10401708370692526999699168797492240T^8 - 680407910276562698898410450598391200T^7 + 10353906760921511571152027566046979840T^6 - 64341197016007520540432295631610298720T^5 + 151935569584040641348413346807219047840T^4 - 98223779992621049253987068203224001200T^3 + 461589258984786807406908653536129872360T^2 + 245414512415050160861358750620214278280T + 41992607521879070487127955356093746390. It is a 2-neighbour of the fibration given in curve#4187 (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 2026-10-27 04:44:02 UTC · history