curve #4272

y² + xy + y = x³ − x² − 7053814015724221775236021940418780x + 245599532869055575551922368319173759212652499490847

a-invariants: [1, -1, 1, -7053814015724221775236021940418780, 245599532869055575551922368319173759212652499490847]
c-invariants: [338583072754762645211329053140101425, -212197996398864015753237098831334224508751020429635625]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 10933472300508705020969962539660253800362336377452333549387041584971844109759650
discriminant (Δ): -3595680007295395624750938366224957142376166944416579039580816499198390200959901134874968447581696000000
Faltings height: 18.4210
naive height: 245.5787
primes of bad reduction: 2, 5, 7, 17, 23, 29, 43, 71, 191, 1087, 8627, 1437607, 42526165861297369, 8240800266961743014968600626321223301
regulator: 19760488592436140822249.3236065392911275978157971415471432058610225474608853392097006
submitted by: apkorte
submitted at: 2026-10-16 03:23:49 UTC
last updated: 2026-10-16 03:23:49 UTC

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

(693862462675865581/9, -341546829692943720365030185/27)
(-89380448766701771, -12728876185521812299879515)
(51870854084096229, -4390346637698918682065515)
(-24953855568982491, -20151445350974727353884155)
(33093484988699009, 6957551715676731406915345)
(-83447435193850091, 15910315523313294744467845)
(-46818918911689771, 21753709733674927632493285)
(2006031825841052919, -2838784221617487582238158685)
(65293144503873909, 7961860546768649234662245)
(-68981667559012075, -20098163041208602359848059)
(1378249379378273221/49, 2858374545467490833441286355/343)
(-52103707986155371, -21718159205334573435621115)
(-22882925517906091, 19875343698327007334262245)
(-53815398240240371, 21664457673428005413011485)
(215544383950286629, 93483946590269046707868885)
(-11593079508880679411/121, 8617689674971585939220265695/1331)
(522514494554858869, 373118636011823724915241765)
(47656614590435369, 4204057068798243807426265)
(23298593599274525, 9690337101548445848677741)
(73313215004942479, 11068316594066765665459285)
(154137361938773759, -53107334747958212758334155)
(1378424279795865862021/18769, 28595856732894822982259296227485/2571353)
(-93961332960265771, -8878393082292608415762715)

Commentary

Specialization at T = 39/1 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -9801785533668367408277499T^8 - 121843241966486333706286824T^7 - 383434732762042957835672232T^6 - 85754468638713501529031952T^5 - 584201627367901316547607440T^4 + 1235140044591109335942134016T^3 + 994771523582906627448194816T^2 - 550790635100880788122728032T - 647714191600162651688347312, A6 = 12142329458953725303990225302566830294T^12 + 247791101750863056653001398463488667192T^11 + 1886090538602583189200926091842195641480T^10 + 5912076790668428510577901681319655809040T^9 + 1282952245208814516295621371483268940160T^8 - 25791045529140834460411754210526102555072T^7 + 619130399100809058015628676691838434624T^6 + 31601519751505535243254509940106610677280T^5 + 10405299225003196422075679041028274690640T^4 - 27172444692997266799042742904489658895040T^3 - 13432438015701644454859263568215615214016T^2 + 8212424024664720375926323815936672405632T + 6370397708074310285430666560056789129600. 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 03:23:49 UTC · history