curve #4235

y² + xy = x³ − 2493926159623851151977372735463463x + 47900036253406741991315607254035780604713197945417

a-invariants: [1, 0, 0, -2493926159623851151977372735463463, 47900036253406741991315607254035780604713197945417]
c-invariants: [119708455661944855294913891302246225, -41385631322943425260059368160404197384843039978209625]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 3331430924370683245282024478207025284146188704187511096808570405072532150
discriminant (Δ): 1542464776091087046121923291078972795031204408049675787664672146050531971195974675289692572160000000
Faltings height: 17.9569
naive height: 242.3111
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 4007, 10133159, 89214652883359, 4261473342511522428524689791491525251
regulator: 340664803994618037017.9603454556256824961912795795508690429825793307462629957218001
submitted by: apkorte
submitted at: 2026-10-15 04:29:02 UTC
last updated: 2026-10-15 04:29:02 UTC

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

(30602031805735222, 489032171040097576414789)
(27329072070803062, 393492189535572497118469)
(29654974200898606, -147727644564549969050411)
(96726946759345462, 26676907111321890224550469)
(28018361374666222, 139735988882568341531989)
(29832798643317922, -224292304360928760326111)
(-55940333126186378, -3515088089462569311337211)
(30489961063029622, 452706080843962774515589)
(26607604723802662, -616342286983713576937931)
(11751473334325648, 4496173275815234585954557)
(24254037537288118, -1296108551754224462569595)
(32671689087427222, -1137673096478350444926011)
(26224688528551072, -730217109784475203672061)
(24124492709679922, -1332506678519533781295311)
(15001855577371222, 3723273935653963709086789)
(1370165217550280278/49, -56917200967190813335629773/343)
(28173078516553348, -7805016586202744347355)
(36722522590856662, 2416340548574785699415269)
(2781430110486667318/289, -24461193711288911404653963403/4913)
(17717765115390718, 3045518831174127419232205)
(12631188460870462, 4291159762675004786700469)
(80792722012168726, 19333404524551404553014469)

Commentary

Specialization at T = 1135/404 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -11140892170449903078832T^8 - 78517047092297850338880T^7 - 410980750922999091180480T^6 + 1172978174531038105951360T^5 + 202359376870041611948760T^4 + 727464198947794396860360T^3 - 533980972281075528757060T^2 - 2994527120513758550219760T - 1424568869019231533236800, A6 = 466667176236662985209696435215744T^12 + 4902108138265145170676233713437440T^11 + 30904617405388839778675659644720640T^10 - 2235477311686016201360264232122880T^9 - 1928619458298746783809833278130720T^8 - 1321657250248806493665223528336049184T^7 + 1152166039831051025589342359093195760T^6 + 3336273056288691191096244581228211360T^5 - 1565104306692700007697832976891694060T^4 - 3952505302344716290555441880536817840T^3 + 26850110070164475305157052575243376T^2 + 1903147075480838773866179023498043520T + 656146040532506595230072169941913600. 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 2026-10-15 04:29:02 UTC · history