curve #4271

y² + xy = x³ − 28654358747729045274796441196339917130x + 64434964588710339099272512212386122044622467244731421476

a-invariants: [1, 0, 0, -28654358747729045274796441196339917130, 64434964588710339099272512212386122044622467244731421476]
c-invariants: [1375409219890994173190229177424316022241, -55671809404645732983834564381338100706339155465584422188625]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 142812942445508923720029062042844910500196247032039142091471085180891501774486242387443517965718
discriminant (Δ): -287858363699738731464006152638883655726913414090345095048838607041309796431041765635891330606758440983096382240768
Faltings height: 20.5083
naive height: 270.5336
primes of bad reduction: 2, 3, 11, 41, 43, 59, 197, 254446487, 604144748177, 686934957628451414261516772217020236649116052535593064108107428873
regulator: 332167987236783109646676482.5860059599029533437128973737758069068768776782492670726612185
submitted by: apkorte
submitted at: 2026-10-16 03:07:03 UTC
last updated: 2026-10-16 03:07:03 UTC

Witness: 25 independent points log in to add more points →

(718506088253287252, 6649628775686730458924639014)
(-3718210555020294740, 10934958125611625288707864846)
(-1496286526370663630, 10196082554770854306302807764)
(-3041056267557444092, 11110838082789717785314226854)
(63369510481644156724/25, 355508100672456584231476814222/125)
(-20256337843882274717/4, -71408619037814654730912719293/8)
(73904629755869640376, -633723931374378802837696694042)
(-5489044628822350700, -7505813709433337407926976826)
(5004924827877463300, -6811151086166219919359398826)
(6788730079381200148, 13519596279765745727832292678)
(18085309082078459551/4, 41804124453258986495645372393/8)
(185077141972999979659/100, 4212106980503335054394183850403/1000)
(4830836265454637746, 6224754914876327094190999546)
(-55507599207170983664/9, 69154101404401936772652968110/27)
(2731721450822745634, 2558157307278100583703570340)
(4837129550609654320, 6245671839119281444718421766)
(-152196694060550899364/49, 3811331546284704800648351891698/343)
(2084528997288245398, 3709709654524485032825769178)
(-142979935085624122376/25, -802772721113084811426778924978/125)
(-13696826060439552793004/2209, -200346901620949668166405925635270/103823)
(7374022972574736148, 15940781721682009125957835174)
(4217391092928645178, -4312827381172778823156138836)
(10558841286668419888, 30644316923766341920858358218)
(369140023761405298270, 7091552368495845524654203808314)
(7688962410127709524, -17282512381147870634996431994)

Commentary

Specialization at T = 19232/633 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 03:07:03 UTC · history