curve #5971

y² + xy = x³ − 5350562664301241326494407752663x + 3622967805182389027179049763745127059952589417

a-invariants: [1, 0, 0, -5350562664301241326494407752663, 3622967805182389027179049763745127059952589417]
c-invariants: [256827007886459583671731572127825, -3130244183677584504723210825565165287396395448025]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 42462656859705214487368244037209023330788587592623444561992701603383470131697815150
discriminant (Δ): 4133049508222893213435278258425610470437878361385466937790212346108761346457425846602430080000
Faltings height: 16.6740
naive height: 223.8779
primes of bad reduction: 2, 3, 5, 19, 61, 347, 292292513, 3996185426561705062200990191, 602614630677817741469201907929666439239
regulator: 333431942141516808309305.403835880982703623851678565164843005352255528541419414916176
submitted by: apkorte
submitted at: 2026-10-27 10:32:34 UTC
last updated: 2026-10-27 10:32:34 UTC

Witness: 23 independent points log in to add more points →

(339700374737822, 42948571920081271667549)
(13009057013295722, -1461373435401486751411651)
(-645354924680018, 82505760297230764266469)
(5302593975713222, -352628430025039628326651)
(5022164062820252, -321591744571400566331041)
(22824986172440518/9, 2154125463248837819833303/27)
(-2060354450405698, 76815977161811281565309)
(-1432459708415548, 91367996731453959660359)
(14895114232199486/25, 3178488856542344645413969/125)
(421851326444966, -37959161092800179215771)
(746274586288286, 6752658823095312480005)
(2031002349424166, -33671759629035862759195)
(637386142497662, -21714937582153128328891)
(33746214765025718/49, -5580451333145011713828733/343)
(3628694128550402, -178851772945559518519411)
(749742153022708, -5732841395371276743275)
(10377238689957242, 1032275861160629323297769)
(-1976986567723558, 80460885830291047976669)
(319827825473116742, -180868540273136292559940731)
(66650508241291678/9, -16426686039999508973966297/27)
(29544973078776947/4, -4846908509811266575546385/8)
(-81309898427164642/169, 171392469529067555012389553/2197)
(277262026280539382/121, 77540795543482242482510239/1331)

Commentary

Specialization at T = 153/79 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -46118133600050365068267T^8 - 175585131457438249904328T^7 + 186310120046915943350856T^6 + 1086331850638516901792016T^5 - 496083140846252073481680T^4 + 514411161151166512216704T^3 - 1349221054433071325320224T^2 - 2378315976590397135297312T - 727839380732795443417392, A6 = 3720542197990934008634945538033126T^12 + 21526619776657745210072107363777176T^11 + 8750301169726091012404314376074744T^10 - 195259865254986340553776941743082000T^9 - 376011441894779917411915234538303040T^8 + 641327438884246873021760128611712704T^7 + 2168837359823326715713321889328303264T^6 - 1791411665482428423973793516575869984T^5 - 3162380047574807481299569084055523120T^4 + 550294990071515361307367747441559360T^3 + 2282676589946398330859127503834660544T^2 + 1241084125645116952889139651480184704T + 235518724223176451772618983340821376. 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 10:32:35 UTC · history