curve #6010

y² = x³ + x² − 10413891369051344192860986017767509100x + 13066492955225037233006841802326122742228921126816922064

a-invariants: [0, 1, 0, -10413891369051344192860986017767509100, 13066492955225037233006841802326122742228921126816922064]
c-invariants: [499866785714464521257327328852840436816, -11289449913314432172317112031496557176829751826686863284160]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 101298815660336815258920678759142994335374832436821638300165912738691271040045037122907378056
discriminant (Δ): -1476599214515086734132199996584438996538127225721276031022080376691143978543899074047222307732312328383128064768
Faltings height: 20.1438
naive height: 267.3424
primes of bad reduction: 2, 3, 11, 19, 23, 37, 47, 67, 728590298213, 10343350349138373498601150036417708283754308390030731900138015826545993
regulator: 72560762634806271024868865.60835799864447570109173142111537959319545338105249237053960630
submitted by: apkorte
submitted at: 2026-10-28 00:47:56 UTC
last updated: 2026-10-28 00:47:56 UTC

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

(1594677472597646254, 717602819334212449989386838)
(2199773354854754096, -896103594871890457588742196)
(20012186283824371582, 88426773161006174101931980134)
(1893915200219673746, -369835412483121826836171054)
(-1275309403449515506, -4926788546460193708983864522)
(1809488528181133100, 383916020891967513666120492)
(2441519723720151980, 1481457447218267731873612908)
(-1865959442841642814, -5099165190651126745061239554)
(2310643003106364515, -1157754283230906504168074892)
(1960187554042094366, 430128392614416988228321146)
(1707359461875034322, 513147297524978796307775214)
(5168502548058188519, 9864615599461605770608324878)
(5423342722103580806, 10775121908271273507980052654)
(-2119861576550953396, -5061248798472481511053180188)
(-1742824700079106672, -5091400416262145189298944580)
(6211397778723272150, 13712274824943258955989768258)
(2555448088240062734, -1772629395230515618890622518)
(17190183782976406114/9, -10240166875286966028269161958/27)
(1870920800849881106, 363035229566902303335755346)
(-1597253763344564087890/961, 151271674546792333693814118780822/29791)
(2840261385920511062, -2530009625970906442605356994)
(2094123947223609206, 664823887029969199444301346)
(-31951598819778137345/9, 62112122971060043051669136734/27)
(1629091257976477772, -651783163184244294235140636)
(2820581157815092202, 2476469833429387203585569226)

Commentary

Specialization at T = 271/34 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where A4 = -80918153007142747807152T^8 + 354290935210738441284960T^7 + 117546745373133609199680T^6 - 1321723369189316995545600T^5 - 1222018032904985784972240T^4 - 1406850947577826580594160T^3 - 1254988108939355569153800T^2 + 805670525674002972117960T - 100416874821843548929275, A6 = 8831810495179731787950781376781696T^12 - 58200269129275243642314244480725120T^11 + 52846203831582430953753256517509440T^10 + 303184046482963991294863102867915200T^9 - 457715356448899454589683173529277360T^8 - 744423311865278126846298019797769248T^7 + 1833256433177951161172716555006515840T^6 + 2794775310889560685872413753874966720T^5 - 133387102323757291083447813526175040T^4 - 778071888245494603901822019677467920T^3 + 439063195335888681561403448200034856T^2 - 128557299724393005377297960904774360T + 14412007642333428007694069449224150. 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 2026-10-28 00:47:57 UTC · history