curve #857

y² + xy + y = x³ + x² − 14756532661864741921538839571827828180694806729493875x + 688733478270817475541584948997649791810530449898809066894953305628342459021617

a-invariants

[1, 1, 1, -14756532661864741921538839571827828180694806729493875, 688733478270817475541584948997649791810530449898809066894953305628342459021617]

rank (lower bound)

≥ 29

torsion subgroup

trivial

conductor (N)

454163780659688159010240261282587950444891915505897227872200070071046313291350152159738966484583252411233415002164998443638863915530

discriminant (Δ)

731159859436630242349860454634359655904699957968233380308809025955222115215589814758146835703588141688160008392413394993592757723028833919485527121920000000

Faltings height

28.7936

naive height

371.9842

primes of bad reduction

2, 3, 5, 7, 13, 19, 73, 83, 101, 709, 9021275403366599, 2236967867397155328119243200592790749607594918740659114201278788059952120886423677018923448040177127151

regulator

777148080130215466899495878570792192.196519265702779342871125712396693319168283971469617689747482

submitted by

thomas

submitted at

2026-09-23 04:39:17 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-23 06:56:46 UTC

Witness: 29 independent points log in to add more points →

(-75413343977389473564010333, 1171616088825289111541930878065118922286)
(6264866532115101045459933967/81, 73203434797901363263721469512083695447914/729)
(35224212579664603919110091203/529, -451399756446181421285261237039426068043358/12167)
(628117774652598584282287250987/10201, -119990559008377436289888951570494905487816394/1030301)
(19419239271196703041998165713/64, 2515933862301781686887132193054518949977757/512)
(-35158527015605917446538413, 1078930582112627005546728368240551168606)
(644959008210501845125452707, 16107648563080951724820570829587673061166)
(74508318951819247547498147, -53676748528142652727528922833555720914)
(73028351652118078467281927, -23635900556969908671224837866449980574)
(-78338547875423538467726365, -1167895879015471836846362498068726552722)
(930808656378637545720375418423/12769, 28784215370303532130118108310103532005426262/1442897)
(5374460021966674410907672147/81, 30317370537081311930258897924043902670974/729)
(2369278116506768628636142207/36, -11057217541974634574441478716012513665379/216)
(559671161222975632735160867, -12951321855992357101113424341620456742354)
(-8543714024696707496692813, -902322239836265424976855670544823062594)
(39008724774155676586716563699/625, 1627819911824714204344370229674896377257118/15625)
(84308121883126724639081721144584787/1535934481, -12644294642930401616427668794154129118082302929912254/60194808244871)
(5909359936674866795193482161/81, 15783146329472718430765906218203544379456/729)
(241411647768098854471821282744963443/3729178489, -15690589685647599618509103437859589065094302654705222/227729742787763)
(343171887223847005713936578707/961, 190653671759026466176736871211333462723516146/29791)
(73823160832040279464756835, 41053833848001636462020145041193770478)
(367823360073644318926779772597043/4986289, -444899019481261590234720804176449014347462102178/11134383337)
(34486742733629211443146201071547/808201, 268687981150752024282993860987533564045151889114/726572699)
(75212204038085437083658067, 65789188966935051120392080507499566686)
(50246033823444189997466331776920067/686911681, 479766151860747777622732103847411575285426678563854/18003268247329)
(76599677015376169686126587, -88528261781931199728772841951100086394)
(12534075760722221713517119843/169, -104394994961800483543688983075173931271058/2197)
(78397545984295248753746587, 117058039004618853307785964158272479806)
(-1577319947438434057483654094533/14641, -1796283618759679999516287086151214576849753434/1771561)

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