curve #4180

y² + xy + y = x³ − x² − 356090127231242115564406287506327x + 2866554315295012188481604526135980412403718460079

a-invariants: [1, -1, 1, -356090127231242115564406287506327, 2866554315295012188481604526135980412403718460079]
c-invariants: [17092326107099621547091501800303681, -2476702928414890453932638828633190114405054648141921]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 128193526569028459166374822630126753659900004577719871983094272591432307462010
discriminant (Δ): -660055060273705897610566351284372804613153197366972995183781092154096453883865283195084152832000000
Faltings height: 17.6945
naive height: 236.6775
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 37, 3697, 23663407, 198249931250467, 96409577270404055114686836277336024794067823
regulator: 10516418279746969986848.6665837235103133454345870996947836330288298678164699601154561
submitted by: apkorte
submitted at: 2026-10-11 23:00:07 UTC
last updated: 2026-10-11 23:00:07 UTC

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

(10654893080705287, -531100211467447987451794)
(19748893291391587, 1880587074382947880590206)
(108194569281523447/9, 15348038279585522255067226/27)
(17495333288726707, 1411285738658064619863806)
(-20705463468839813, 1167389173052977933587206)
(-13122047917405013, 2297765221624098152439806)
(8530900525691587, -670545997197072668009794)
(-21012439805623853, 1035088927064065441681406)
(60191676162828547, 14124787240803003641938046)
(10452695869389187, 535258860321913917102206)
(11135220629968387, 531133817472486535374206)
(32394529927353007, 5032512492451564393286306)
(9321858570350467, -597640779022162460751874)
(23133040030709263/4, -8002479617216468849683877/8)
(107582328711330887, 34780897636981722437124206)
(10576632046654087, 532425150388326750468206)
(4051646839459087, 1220784201486365605548206)
(1169896948174590043/121, -761701870176456496763575142/1331)
(1977908286526155803/169, -1207570290367833695061177418/2197)
(809419343305987, 1605882429060876196743806)
(8643411434619602467/841, 13188519513292426483529645134/24389)
(454543597174442917, -306193087871887132977447094)
(-21349183474536413, -859123920640160914097794)

Commentary

Specialization at T = -29/96 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -6490299971789709696747T^8 - 122171843220159373604496T^7 - 507701052166171299717636T^6 + 1050551383991494297232448T^5 - 1535498496547466925264930T^4 + 2182465350304253473007088T^3 + 607653974805791446562364T^2 - 1786833563705739775293696T - 740288204118893840150907, A6 = 301250623701631052997714872966886T^12 + 5692055598226519603489686839092272T^11 + 35336888180450378093651037180173676T^10 + 117205369002705497808649348735832160T^9 + 178701654202460409889459615037185530T^8 - 1878311915395808086196582818654473120T^7 + 1979838463701093911232660031750203240T^6 + 1153701456205174682023944677522475360T^5 - 1378053281785187552098745035715906070T^4 - 1354630698351600468048880643107976400T^3 + 348278788638126247479431301650393100T^2 + 854663677517493018162704474754865920T + 245812677073686954605814171513689430. 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 2026-10-11 23:00:07 UTC · history