curve #4209

y² + xy = x³ − 3085484064641879854013532386492189836250494832403019772295x + 71163454605053788961138963180946822342777224886132148309377588615094552140649626278025

a-invariants: [1, 0, 0, -3085484064641879854013532386492189836250494832403019772295, 71163454605053788961138963180946822342777224886132148309377588615094552140649626278025]
c-invariants: [148103235102810232992649554551625112140023751955344949070161, -61485224778766473662424064188560209356813737651107150471129674231651728677454294527818841]
rank (lower bound): ≥ 30
torsion subgroup: trivial
conductor (N): 28850698960424975792029590160908875160408996461393541447176462119216042862293632065627211838117534261663670289877075796792267909081504910
discriminant (Δ): -307784923608426635320722176569555854603229892136853011110240303848684705489741311137352566071453207450843347477183655183527969689667266954791058859486164184811158466560000000
Faltings height: 32.0311
naive height: 408.8874
primes of bad reduction: 2, 3, 5, 7, 11, 17, 31, 59, 67, 167, 193, 5630105814551309863194864357089694629, 33038216314247152083833824757800519093792971773600825025064433836490620691006650888369
regulator: 330284671568083991441707577494511243.30696987108145321712772875674316800672283891177847274939198036
submitted by: FNacc
submitted at: 2026-10-14 00:22:50 UTC
last updated: 2026-10-14 00:22:50 UTC

Witness: 30 independent points log in to add more points →

(18489650183734796288689239990, 4520501949171783830206522816506396845092005)
(-4963206632648009483779552087110/121, -15100343810933364259437862351416981906625578345/1331)
(3622355193485717337929183652, -7748178234754684473290166070434411712507689)
(2066828893780810958897343313995990/39601, -56988081127189065936748726901514070031319009337005/7880599)
(137156603349839026616778846675/4, -19066457957797949400980107110856539436058685/8)
(460384315282160383307031625318470/5929, 7904241353512301089540502092108406853675443957865/456533)
(1845534611197160283583579940950/81, -2596982601309168321390914522107516216551669155/729)
(134937283737736978796132353890, 45953935598801050973872731930686163676455105)
(1260137433135640949672180057190, 1413228101695748004135392016676194575166952005)
(49685685115507605177632620470, -6365256580159492828651138846750492385264475)
(40222320262485277237274234190, -3482988055753840457512832693502158534376195)
(61342529344476103044964180845720/2209, -272632221185756086123300275195922179238707514155/103823)
(-60149826673901004996317758410, -6255606622151283537548322655408191574421595)
(6129495680212351933044911030, -7244396655955680890912873055164502658105435)
(-833272502310070078424797655635/16, -609177379271456159478023929302345804139912405/64)
(-342083595962053085517958203756/25, -1315895655989399989445749810961195878734114357/125)
(53260006838813154137351825030, -7609822362886309233760036057964578479705595)
(1948053035800398043564149995257498770/38651089, 1588187610018214484120684981754237827064042138132150645/240293820313)
(391967647957856093513079941844390/361, 7750297519059315592086968174496786067889647406295/6859)
(423295208494918320533897484625590, 8708934515368197440740991031299280634150733860005)
(1308171889845029749773604929587190/36481, 17942791092152863991311861362775272457042309067355/6967871)
(3104847477345421014895018850190, 5470051427871158222285823410566870924783487805)
(5462100396938263700725942524630/169, 5010662325279499536326741810752861150623026985/2197)
(24320122685309002804704850230, 3241710485181764911336347429006601205635365)
(7399615598318304824469808664646/25, 19799069681226785001242342735644156749824048529/125)
(211034511936419537251243700790, 93907194094438425013754785107554952042695205)
(33185757572907562677764929590, 2305779894723033375563646981727306695940005)
(4886768655816942315996430551990, 10802014191123584894878523336824360716900868005)
(-36006283595014856025997613610, 11643875385049337767374606989133876110716005)
(248148747499483274765105764350, 120772342082363315861240964308043811118197845)

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