curve #2186

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y² + xy = x³ − 71813598680248384341084284771096244120x + 234238430204114181370252185964622864112853337413958990400

a-invariants: [1, 0, 0, -71813598680248384341084284771096244120, 234238430204114181370252185964622864112853337413958990400]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ × ℤ/8ℤ
conductor (N): 13009810086729290992290
discriminant (Δ): 30141830225172216740152510329492050833160247441532470935014105187061096434307850380356942141856153600000000
Faltings height: 20.1757
naive height: 273.1150
primes of bad reduction: 2, 3, 5, 7, 37, 41, 61, 82207, 87697, 92863
regulator: 648535.18215260367820925454406977072541214509
submitted by: Tom Fisher
submitted at: 2026-09-26 17:39:51 UTC
last updated: 2026-09-26 17:39:51 UTC

Witness: 3 independent points log in to add more points →

(-7598178717919023024, -18472425486358616520979672728)
(33107518580292000780450278609381760/14540188262786209, 15927641392698942846938889124910803029530512142688360/1753295202855635392342927)
(7169614805036899134023153141153196990357085683379828079536284782724510447167404998785943150358684964695631101220588376852024665148311485010003860968019856902280419518976437129290556590850089313845905282923159362906272145175547483113113434925455756281408466044338573822926291239208328847688535383453315074754703503870640/1465359420784521666593500820186204902125635491276180880963545711302043111411259819241169254910223808848889344495128879930179197148977231505337456585096880170139704564540736082118825425137138526515935888308875853360197443623012513977237432276380370788511664976968599329069550249184249712258350070642441, 19051956978166304106636802088573262331706506956418638803175373798556054649348985168693779338395235722904756284015499488031502683118919663470431161407332137187392725048764386690185043801909176315114878304901822975091839245263866757314049094445831863647075924599883755663792503434157502890358923060353677352313056482590085663504087943683812759287294776392715727203942616363548265111224826270514490852966164516961806584087137112284509501589805618445202272808534774622659622920/1773847340225989184102723728772942402900199513803075093987876252175123264698561137394357540918127507323976256544741489648744105911018442006995904686890599231975654318272238013197969903824088190545915378633831874382204500133419698516514042948690030046152357181320303410975906192377697645691396346590434092119917768862526923404926749336715099054169544666172146667440727804515415014025670218232007675920182529293676086193703390984841249073895986347173989)

Commentary

Found by Rathbun (2022)

last edited by Tom Fisher at 2026-09-26 17:39:51 UTC · history