curve #4221

y² + xy + y = x³ + x² − 59571686638697407743468933983547849913750x + 5493310309765056002343723952555166972425698346785431891678867

a-invariants: [1, 1, 1, -59571686638697407743468933983547849913750, 5493310309765056002343723952555166972425698346785431891678867]
c-invariants: [2859440958657475571686508831210296795860001, -4746220107637008386046423302197595330963452187856690380379491249]
rank (lower bound): ≥ 26
torsion subgroup: trivial
conductor (N): 12155956057816879775291315106720585954250504860071696639621525101430862242096977971482880863570610
discriminant (Δ): 493828221918437963313966250978932684552232315170943953955173177196344015469135302530975649727286178931928374742039040000000
Faltings height: 22.3322
naive height: 293.2776
primes of bad reduction: 2, 3, 5, 7, 13, 17, 23, 37, 53, 73, 137, 108447949, 581664289, 1391773349, 7906300695696131, 77640198694120627, 828825525916519930703423
regulator: 739569374363533190534254173.407444773379971979512593765532871464421318525594844622554364101
submitted by: apkorte
submitted at: 2026-10-14 09:37:01 UTC
last updated: 2026-10-14 09:37:01 UTC

Witness: 26 independent points log in to add more points →

(309146429441221758937, 4077073540700166220890968235031)
(249050171898397561697, 2470739669736873353691490630191)
(81356811069207926257, 1088688637662779071233384805551)
(76107956786838251137, -1183334395326554726849514218769)
(175576624067202781337, -668162979975660701945683654569)
(124431745452084988937, -85500519467475337014679953969)
(211309388432013553505, -1529891534352181666760702964177)
(-236692163658198173063, 2516587758179129705861618615031)
(-126082020752649311163, 3316616334958893129957410277931)
(252445072221077277937, 2557866496892491791575403258031)
(176565438477501605537, -692458950274858557616290449169)
(7807875129385929763937, -689587692278799098522826931509969)
(157581890558304548705, 137742668118975682186499647023)
(161183033535454589937, -280883549939933581290970103969)
(-203245988488922967583, 3033997686653066260358199700911)
(48768793526901447784913/49, 10471975384161998476471341550584633/343)
(57783700819653412319337/361, -1665672688929770421400084638384251/6859)
(13859370346787831115437, 1631354489776667614398628541114031)
(-142174065302646783583, -3330019324993493117712013555089)
(90789152268818985077, -912793743314758601087935105329)
(172506920057805699917, 591906259186978346159384919171)
(4227218139790884894361/25, -63100242862584495891004118255949/125)
(235300596497616163337, -2122215107919815360002859255769)
(662636837703332856187, 16030424849657823328494693456531)
(19162157382388080677153/169, -973790700969194026545112487949693/2197)
(165230606069182087257, -401535189745173477659778558449)

Commentary

Specialization at T = 1050/61 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -9474380172051378777648T^8 + 143016499424841795151776T^7 - 978479134779758056367040T^6 + 1264156022749661958710016T^5 - 1005963702085547848031760T^4 - 1438422958271309512308048T^3 + 2172965217056038418681880T^2 + 1222317589612619797944408T - 1528603489372304441232027, A6 = 356599201940399174539163794687872T^12 - 8095747446508036482913060524583296T^11 + 84031424784510859456436955014577216T^10 - 395756683659861170554655745272982720T^9 + 1041818162127380294010702794653004496T^8 - 91848543443162490589835301473411808T^7 - 1400855594533976892297125576033331840T^6 - 623560771669003657652924198286452928T^5 + 1801926810925269265387912620705751296T^4 + 1152132793779120258533624208322687440T^3 - 1448000646205536545177160777048329016T^2 - 809134531551941303559229668814825224T + 723128597502954965566093502187696054. It is a 2-neighbour of the fibration given in curve#4205 (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 2026-10-14 09:37:01 UTC · history