curve #585

y² + xy = x³ − 6226812508901378634130301390817577706414382815550995x + 186115121994069596045152284611073457436562770843554482219936842626375765781025

a-invariants: [1, 0, 0, -6226812508901378634130301390817577706414382815550995, 186115121994069596045152284611073457436562770843554482219936842626375765781025]
c-invariants: [298887000427266174438254466759243729907890375146447761, -160803465402876130983011574352297967866089495666212772776891026891024224354477241]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 143228513361520618192311577687396048144845234090502739942156915264171700184216521077335296006029488761851148710920809341970
discriminant (Δ): 487759920303004769902669782582144331146824895166457106854414059127033671216746199771686619834309882014568860918566666085910490322658870171171267411200000000
Faltings height: 28.6687
naive height: 369.3957
primes of bad reduction: 2, 3, 5, 11, 13, 19, 23, 29, 31, 47, 59, 79, 97, 911, 2985901, 4708681, 66796733, 22850413867511, 21714059214839599, 9421043469351579162074658386462269775293343803246867
regulator: 17162450000456538701978772279635.709257477653837907059262375401998108293208433592297444562434
submitted by: FNacc
submitted at: 2026-09-05 11:54:18 UTC
last updated: 2026-09-05 11:54:18 UTC

Witness: 27 independent points log in to add more points →

(-89829788701759738960759130, -143515473115234264911835051734444552695)
(-33816722241414799870516010, -598342458985646564665873350588674926295)
(1208909085353402873555597640, 41945585694451818672440943501421940072055)
(-64420195218661200014614910, -565602569015569512956896684519126568195)
(299415594005890159783534528/9, 3391703913610960022060242537710715312117/27)
(-2695049072200064385552368, 450418818820125352588897327685793218431)
(56676853956186223429261750, -123531765052525752331324557548510093855)
(-90894441627937477669352510, 33854305550027490199437745637639752705)
(24957927527730737518127286526, -3942853227647721544996455594773945478063143)
(482735188688648105110861690, 10472514956071116689913937335429016599505)
(321477118820008508933825902, -5604302463858690705758134537508010488039)
(123301388883805019564932270, 1137066477469109765526063435249191789305)
(1265430662350965596428443490, -44929492703179423485665305690171273663295)
(153327125855183624872912015/4, -490039574517952977496642574007474526495/8)
(-50101470834798222874496810, 610184407001087038759280537615453790505)
(39663793524710721073933240, -39189940224321326571820274451377835545)
(57259205454901550343372430, -131543784570112442590125867818647832135)
(17761146883167940200924640, -284820456247478132420412365966637344945)
(-41998397756074295624913810, -611188800001093359372191775756771627495)
(32050858996129314586237150, -139516735773180946079574715722426635255)
(40605353519986067169589390, -14936259338734496266054199209505274695)
(63974462266056337359622702, 222684659624200083427316889806741126137)
(50355759302767769526296350, -15699508652058674032056421533192132455)
(25291975065147422642178040, 211673350068789776081759786985275823655)
(318073297942129600420001890, 5512277750690865455481678142164115610305)
(-792783847228002266189989730/9, -6104601925952645171177525563061931147965/27)
(-2963550002771080705217413730/49, -200437723338139086042334622228836231442705/343)

Commentary

Specialization t=-4598/2037 of the Elkies fibration

last edited by FNacc at 2026-09-05 11:54:18 UTC · history