curve #4253

y² + xy = x³ − 29332823325060650230074659797865743555x + 65132098912327848069391029885497504608521560893445497025

a-invariants: [1, 0, 0, -29332823325060650230074659797865743555, 65132098912327848069391029885497504608521560893445497025]
c-invariants: [1407975519602911211043583670297555690641, -56274133460251260734065813100474210798328004117383242965561]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 54275045947234878747477255115920299904009892715164363090991870522074318653843577009641023336490
discriminant (Δ): -217369430954043749029798024293302595528576395190366834142287100502119500730979156510801059456761115990584960000000
Faltings height: 20.4948
naive height: 270.5552
primes of bad reduction: 2, 3, 5, 7, 13, 23, 31, 6713479293533197433, 4153367585118333162290315570397757229716047436670281272008366959420097
regulator: 270279687383918662328170741.6898351820211644842515795654985213954187906682280520242857961
submitted by: apkorte
submitted at: 2026-10-15 19:48:22 UTC
last updated: 2026-10-15 19:48:22 UTC

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

(29179822216090424510, -155095828551921779443322431255)
(-2669663263862820958, 11154097792511857930898562641)
(4680301338572967830, -5510778534338373133419386575)
(-5722421622901595890, 6752765202638050436988844745)
(182483260029464629790, -2464027297902922903327692395335)
(9029414753462891215/4, 25824288151673692768264624985/8)
(37829807705494820270, 230420270663004944289929048585)
(1594072666193706260, -4735408487448792662266307005)
(92064572454860865710/121, 8753718749219145269226524727595/1331)
(7799207529169735310, 17628576095794494822523883945)
(-1069345810300799316850/841, -244345101816663633514055212746995/24389)
(-6288084150024392890, 973908994385251948006459745)
(-6030924427169022490, 4762253476932777795935935745)
(3119210330560031150, -1996278121940916944682110455)
(84446582784164650526/25, 267836187793975377752292983461/125)
(3786962564249327486, 2891184678434425304474792297)
(294318383680705670210/361, -44323875785379893848294376539045/6859)
(-3913316230981181614, 10954086722554167097762669757)
(-1397014537601174938, -10167792121355267295966353383)
(10226654585608588010, 28891255336802560947639444245)
(195377007639450936479928590/29343889, -2041796732445637765955955706152471978815/158955846713)
(3926323385674944934010/1681, 210877220121397602620463242095645/68921)
(4005915932395875710, -3451327218225251234035567255)
(734635657111874646706670/116281, -455533973905590579634959559528024955/39651821)
(23663923139261938970, 112349258459671551972527288645)

Commentary

Specialization at T = -469/370 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -6045653798989791126192T^8 - 62415704538217319360448T^7 - 1599340192212345051878944T^6 + 411285604330977976052256T^5 - 987166911410498278789680T^4 + 1046627474370060703109584T^3 + 194010634771867183414656T^2 - 729479896920873105811452T - 225965831905541594247667, A6 = 163989304706795551092980878190976T^12 + 2463417736840188795628573229020416T^11 + 99064787504796030904023011794321024T^10 + 306093533334403679180135432179319680T^9 + 1458791938240704239622260678224652160T^8 - 2067052078435764908208365147286526272T^7 + 225696680714681572698300960927997248T^6 + 1453625359670481930865233085816499472T^5 - 247522312386237949189300655828535660T^4 - 553746777745423527483394159931837640T^3 + 133678131457935919739674847770663476T^2 + 203231717596297497476228246857348516T + 40705869704413231155701476152234574. 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 19:48:23 UTC · history