curve #1682

y² + xy = x³ − 42345324111085620184700223297082981326689873538x + 3364071280315505150448571574468738762599998567472118156176194943402692

a-invariants: [1, 0, 0, -42345324111085620184700223297082981326689873538, 3364071280315505150448571574468738762599998567472118156176194943402692]
rank (lower bound): ≥ 27
torsion subgroup: trivial
conductor (N): 4158960092540785960398822964998502415831019202640925517818827528346583882085978524868895601057569033333947600550
discriminant (Δ): -29380060272617969719452807715373927965480283664717714736940159591758922179803489777920009707191503228156649673680023732273250937643904000000
Faltings height: 25.6241
naive height: 333.7062
primes of bad reduction: 2, 3, 5, 7, 11, 13, 23, 37, 59, 71, 89, 2377, 4861957, 52826660453184490529025390279973101133203797217780130812237432088688647399873759526261
regulator: 757426742318555778338020560908.77122657122444246594571165261267588055409172974439173718913394
submitted by: Ritik Jain
submitted at: 2026-09-25 04:08:47 UTC
last updated: 2026-09-25 04:08:47 UTC

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

(258026702482646468726330599764/1708249, -45434739069641263933403105201025536069726838/2232681443)
(229368627554746450360076757396/1890625, 9149816628779748063271994644589961846154194/2599609375)
(216562276790624434256510427/676, -2646727011767728881685069071169574416701/17576)
(12929265882509712101288988, -46484281729898789742606681076822158378)
(-542638950477662102326717/4, -650513845847760539132633134957532033/8)
(81506243768499710910582, 21310344281719994798195847333102984)
(152874883769436701802852, 21525157941050878296181328073950574)
(13747189279160064676164385848/289, 1611821006291040101007523052644715296452942/4913)
(2293978291034765454737604/25, 1980723432862716327703870335717697398/125)
(-952570816645619886090366070668/4084441, -194417817714519321163058999515201718735960586/8254655261)
(53143895571486734615163768/49, 380900883790767195075013393993418426262/343)
(317457327733344682152396036/5041, 11014834196922790281231318071315998299330/357911)
(4599982594838707222947744852/52441, 216354958419012794474024869737318824227446/12008989)
(42356892485084629394007744/289, -85957208289821584142259806009880688110/4913)
(-2813711514324297636915653844/11881, -13581359912226136861249023110977927394274/1295029)
(13947730992724769230966836/121, 5074748557659223936643405408921002074/1331)
(614275148262185272805502, 457317465103362322757375961775595424)
(37542202907527679120748, 42746311528761252415238886185456070)
(-16651493311754478229444183342691231889570139531812/90111665965133521237977529, 59750705257130252109243618711229621440316490611650275747272944635825303562/855404495576705105013795867111246000083)
(332473373522701940268974114637372319032/3919364363699929, 4798235040957964286502683536298578972732092656562651411786/245371127939399134957283)
(952071252765520564817229334655138394660157401/9992720362110627993856, 13928962022053255350405791191960917427849053846564667269308604733221/998908253064940708601839223189504)
(-2208768184683282737708380216100561172188/10370615907560841, 55095782660781824765747830153164582918062896450096952005574/1056104333468481375021861)
(106070975227200996166924753131109194559812/2654257478722045681, 180155420613009170593255358965450176241974548273260079828157246/4324287642241503933890256329)
(369488625552911094084077800944526581092/3378665705416681, 1258943675102747168182156837709387181629214336047450786714/196389305984770276060171)
(1423588876315148064456619193290515210308330389906410409412664/6624264695647695078039476833395896209, 1103482785461116372148321223828564399008144109135625563429658377694151011563635385992432310/17049298591663662354566909877209744897308811043549337927)
(176575560540650477568115633132231720272441488705581970131258641194532/1130990418886962607290890713850190378577514041, 898834710585683006216314663083645742759228877616983255933448923928859303201351528454083992896078331786/38035435109290740640620582526870857525027062300945049339969468152339)
(25092516108957056266532374025546160678/238972116790729, 32082204405897470973297913674691205944714980132162055128/3694203758191528579067)

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