curve #4236

y² + xy + y = x³ − 55954035423095117031205350678587424426x + 158494191015267709069669837780824619213169697758645314648

a-invariants: [1, 0, 1, -55954035423095117031205350678587424426, 158494191015267709069669837780824619213169697758645314648]
c-invariants: [2685793700308565617497856832572196372425, -136938981037191300640223430393095319426425404112327846414725]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 31806969506801151852156648388722049730308181927719591097449259312127947925467681356373925674035950
discriminant (Δ): 359754377688695765544584966027922965049761109507365404128287453939552552718903079704976545973219710860523009292500
Faltings height: 20.5837
naive height: 272.3664
primes of bad reduction: 2, 3, 5, 7, 13, 101, 151, 448175827959916799, 2276177021370588202448759, 149774147356843215069689409757982345350847837533
regulator: 886518569677594013036404757.7576377162040506554848688466363176115146335938292095890962329
submitted by: apkorte
submitted at: 2026-10-15 04:40:46 UTC
last updated: 2026-10-15 04:40:46 UTC

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

(24974263163442869543/4, 57981448963765348917127677993/8)
(4759746197858435282, 10437094139228094426979641)
(4934121026525872862, 1591885663812819312482430636)
(547091065515200098523/4, -12777692744432260034523202706817/8)
(-1789975632834945148, 15903315573526485962625421131)
(25018057848866236967/4, 58350067947972302953232897481/8)
(26442670344609186257, 131026817584393758848186994651)
(23124912488207197502, -105975830757326009789243590059)
(13380127126348754543, -42488078718555495391734387633)
(96875812084255972757/16, 413149523824358915925297657759/64)
(49203023012114864027, -341354403423026298085020054519)
(-8132374798040366284, 8700261127048716610518526416)
(10555109984431371182, -27273460672281451610101955124)
(1729269600408284828216/361, -4225058786879159960631020937672/6859)
(638947973521285962671/121, 4265515750278886356778223871027/1331)
(246798900468143439527/4, 3849888262049326285572853630641/8)
(4760361837356792711, -86655117727875873899565183)
(-2253530389445322034, -16527068198949592767844619508)
(4924128132236186897, -1537814145745539382093377429)
(-192991568056801360696/25, 1427416402565561241623202339123/125)
(43809126068561862143, -285984970236182396462905501311)
(450160966379818932233/64, -5440565565899095046836340873553/512)
(834511668837892619788/9, 24031132857856935108002140145057/27)
(-5138378672818647115663/3721, -3465670406407100220887043959755389/226981)
(3217671585849796952, 3430212191842674479612298216)

Commentary

Specialization at T = 970/479 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 04:40:46 UTC · history