curve #4577

y² + xy + y = x³ − 32758725446042024501993255323248138x + 2309347087879937344235044396983813139764936209623888

a-invariants: [1, 0, 1, -32758725446042024501993255323248138, 2309347087879937344235044396983813139764936209623888]
c-invariants: [1572418821410017176095676255515910601, -1995275883928265867777706591109040316900419268388905349]
rank (lower bound): ≥ 24
torsion subgroup: trivial
conductor (N): 325550369236788627718104773220234002353514063832382841479696492039232588130487188906570
discriminant (Δ): -54003948709689594975718186584382938746412566399296563590564349762911200920114290235267682822326194062500
Faltings height: 18.7106
naive height: 250.0608
primes of bad reduction: 2, 3, 5, 7, 13, 23, 30653683, 751048873430637389460141677999796623, 225204387997853269648945329203998914787
regulator: 4556079942979910750567422.5091548078411392847674409710827579623597543492307453628257849
submitted by: apkorte
submitted at: 2026-10-18 07:02:56 UTC
last updated: 2026-10-18 07:02:56 UTC

Witness: 24 independent points log in to add more points →

(106902393207420104, 5390222208494664227967360)
(207787663084580054, -66886983692513303009904315)
(167257377211172927, -38848922500191713210270043)
(1414858645124344336/9, 872666701801116624774315920/27)
(141061345978707539, 22253957650320747663087765)
(-61478336537351713, -63960428233184311341075483)
(118278696549630379, -9454574148876542915193115)
(-134798396431697131, 65389616747909525165235375)
(88675926764048549, -10086268148367172700657805)
(780602045616884204, 672599078574973479933267960)
(4008389305591631429, -8017139856501324923204941065)
(3000273417723239, 47022217748289536153618115)
(914489542495616204, -858563522867030746618140165)
(101151723352328579, 5540462248150364823907335)
(105004865061047882, -5225654288703944419828788)
(72117771069766079, 17942781210470884752626085)
(156518106947225954, -31880732717786532204392415)
(407913095045419829, -238370626762541571262969665)
(10572204093247454, 44319260229210976673225460)
(41435703794900991821/1849, 3166819862605017967842981171345/79507)
(188604807141536204, -53290725661684957153020165)
(256590543801661264286/2809, 1324662107761421132908625796420/148877)
(17806079592247214982791/139129, -757554502936941201412627837836255/51895117)
(91120348857449954, -8995853513145868955793540)

Commentary

Specialization at T = 682/123 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -485042948797679528112T^8 + 12463578805199058894528T^7 - 230629938307243679610144T^6 - 1586012147509225586376096T^5 - 3868907398105357015696080T^4 - 6794745991541079206248224T^3 - 4791118686138149116505904T^2 + 6061871481875161826582772T - 1374235222299991775805627, A6 = 3820001086939854414872700973440T^12 - 155249980524531925200471635032320T^11 + 5279785372691676756614905714258560T^10 + 22439305421110286908829252745667200T^9 + 198311921848923368813090766393603840T^8 + 1955108933617356512048828761030033920T^7 + 6903605561322701279549833535328289920T^6 + 8139034537986456490540355512081445520T^5 - 1317692978359257302156310289643874060T^4 - 2476393922218195353910049397638432280T^3 + 5841882186136964852470754649206854356T^2 - 3899452237007593054578752344925650716T + 735364979113660346489805816953176854. 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 2026-10-18 07:02:56 UTC · history