curve #4083

y² + xy = x³ − 14694422469323189681173353938479126935696990936020730x + 685415727609054669446260830821269920693344762172817946143147166705805190516452

a-invariants: [1, 0, 0, -14694422469323189681173353938479126935696990936020730, 685415727609054669446260830821269920693344762172817946143147166705805190516452]
c-invariants: [705332278527513104696320989046998092913455564928995041, -592199188654223234401569358887575629270319531561796189038176291403999031999707089]
rank (lower bound): ≥ 20
torsion subgroup: ℤ/2ℤ
conductor (N): 735726883900703598396459076522528622439471321942270014656714425868103908609679211118354170
discriminant (Δ): 114833168087293521453512196807849998949185738825459777856164142201856948611578456306799030910498387524627441101713372455060118607309366012308921986900000000
Faltings height: 28.7269
naive height: 371.9715
primes of bad reduction: 2, 3, 5, 7, 11, 13, 31, 43, 79, 109, 113, 127, 149, 157, 293, 487, 499, 863, 1319, 1481, 1489, 4561, 4943, 7079, 11549, 28433, 40879, 108643, 1185883, 2048521, 5719267
regulator: 167757052801705547031091.357799176387758111805133955885504316938189366237747359
submitted by: wgxli
submitted at: 2026-10-09 05:49:50 UTC
last updated: 2026-10-09 05:49:50 UTC

Witness: 20 independent points log in to add more points →

(69021777815921504526642804, 845888790795838950562826092072145958)
(68887001539577295334749954, 7662179814442029880472674237160232138)
(71030048926221358172504744, 5983555350000184373682539778429737378)
(63901109044069226449399344, 85770272661902303339128692721691574678)
(67843683933801808542250344, 27577860722976707479205244628569950178)
(68012461552762923309548094, 24835110534020124107544734631702724728)
(68075386614135742630236744, 23794113537638994503777634506423253378)
(68629277705771751613262004, 13802130497720310833801421907821327558)
(68799359125642106233970994, 10022497792148621744182771872435642378)
(72137263672778641823141094, 28055237831661895005703936461392047728)
(42714989903986559273417844, 368347831268472205873833824230135752678)
(59985283259973456000814494, 140739943279584567746784574142969970378)
(94634772976623914488732392, 377275548105221333961744551794254635190)
(156216687407828702567506344, 1483967528193077514056343083043474402978)
(217906140144407073274103484, 2798261392192359515429477609161670513718)
(271717107060301663814048451/4, 209560924775190039174285675626766889599/8)
(2423457423529632111500366664/25, 51825406163580845836185460206508899108702/125)
(44219060223314944128302949376/729, 2586587843283742599712508147793496710804674/19683)
(363978244410874656846488801184/3721, 97360064842976118296080928123015710517402478/226981)
(87768988176040498160811833664/3481, 118149833129017035143960939306459831627852862/205379)

Commentary

specialization at T = 11549/23696 of the rank-9 fibration of X808 given by: y^2 + (4224T^2 + 29864T - 18691)xy = x(x + (289T + 239)(45 - T)(5329T + 21775)(T + 6))(x + (169T + 874)(4489T + 5852)(7 - T)T).

last edited by wgxli at 2026-10-09 05:49:50 UTC · history