curve #819

y² + xy = x³ − 6495649539452475553983700741659641381973061496445119220x + 6302745660233214380787297860135380626275509448727774485858616327839978267353056912

a-invariants

[1, 0, 0, -6495649539452475553983700741659641381973061496445119220, 6302745660233214380787297860135380626275509448727774485858616327839978267353056912]

rank (lower bound)

≥ 30

torsion subgroup

trivial

conductor (N)

1121230618684897861116825517533995696067699433783988527774196776647498191355638701562523035961442256305324527830432958159280384768330331345790

discriminant (Δ)

379704244571207059987387785369786062433931595790679210524088306746766487422641350730044341568430467874505181881724178475857808060131039602220838195730479803360000000

Faltings height

30.3893

naive height

390.2458

primes of bad reduction

2, 3, 5, 17, 23, 29, 47, 107, 157, 35376078291767284724288792987, 118007052933122043386079701598996446446101278818598100878749033750801221345322480968934502644389402617

regulator

30587907882744214240746023474485941914.803740407837280259134597070654661380677653488869163402110440

submitted by

thomas

submitted at

2026-09-22 00:59:58 UTC

later contributions

primes of bad reduction recorded · thomas · 2026-09-22 02:16:07 UTC

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

(1961949467021616123064104264, 33326192098169370155238984522894385433868)
(1332998788764893979140127264, 3555175412926387168684703270110188545868)
(2733614177381887948226733264, -94728445162885965348935008617480514926132)
(145879048275687442896615564, -73200211032964219745977417419847903892232)
(1209614808059186618478584904, -14675879848352289721316437158350893135092)
(3632800987086740371689697014, 175066561745960599441660758771951365427618)
(1797892762881393381577155528, 20875966739494032760938429582872097222156)
(26725731189891002111086340064, -4349936589977597884693172576138737710738732)
(1194007461809220531163759344, 15783945593554984455417923713893644669508)
(1212321523131202782204246264, 14481274777223897208011260915716986225868)
(96929883622613319656623146089616/27889, -746790510168497001549933489271206192513421149996/4657463)
(-2788903278368220175744631736, -52215379558152773115955662938432464422132)
(4905132504955514436770147817864/2401, -4645318073293951856167821219515422651452931668/117649)
(1595479550989241955083189064, 671263192042386451070572769434880768268)
(1258142427553961947140189384, 11037897030050017180034352242588162789388)
(997539000058965332990160312, 28560760997358122702491968608966674475388)
(757130351338719223193627228168664/2886601, -333243634630273552105285671077415852853463721091068/4904335099)
(83488634888539920709488676521/64, 3585868630789673459990084237752545638977791/512)
(83397417362719111488038388936/49, 4569558922631576287492068648227293807400724/343)
(10186758017401126799947839018954/961, -31647511139694470597867833832770534016484529662/29791)
(372807910797598821154269464, 62713085627344737360885414101635954725068)
(63382205371481970426213534712525141704/47974578961, 54587904055837844780219392938101035402775780838861621588/10507920004406791)
(1316304787443912186579631752, -5761725484937141504371768997001371187060)
(9387445153397640685742912543983869384/4116377281, 15285296858774520440116487443429905352611046976723654772/264102649971679)
(-66531914910881828514385806, 82064725427235794815374222574709746630998)
(1320876893109406706377300464, 5229437749690072375879928198255100363468)
(88197164432868561687996543096/4489, 23634557905755225689703857385132668127174385284/300763)
(29953476239921722758347958440106/29929, -146835466958108249537662934762270174580106824744/5177717)
(1160089949762205054964111813897224/143641, 37742017181357838225594190864366518715582182050052/54439939)
(4483443290832954499743981869676/9409, 52553841548919845297586912163340471472097731864/912673)

Commentary

Specialization at T = 21973/509 of the rank-17 fibration of X1092 given by: y^2 - (Lx+B)y = x(x+D)(x+E), where B = pq(L+p+q) - pE - qD, L = 223949T^2 + 709916T - 527647, p = 672144T^2 - 682426T + 134980, q = 180052T^2 - 495787T - 79089, D = 74671643904T^4 - 215678235850T^3 + 42928783983T^2 + 98361938857T - 32113982430, E = 127560510984T^4 - 427630294148T^3 + 69500394306T^2 + 100280044148T + 23814085350.

last edited by wgxli at 2026-09-22 03:47:59 UTC · history