curve #816

y² + xy = x³ − 408232930787546995215449775725226828441376566694203766920x + 3164819895198251847520301934841034703331069491403820539317437278618088006752094222400

a-invariants: [1, 0, 0, -408232930787546995215449775725226828441376566694203766920, 3164819895198251847520301934841034703331069491403820539317437278618088006752094222400]
rank (lower bound): ≥ 29
torsion subgroup: trivial
conductor (N): 13586644220672721216394186457337751129627290081495005465673933999291044189221572379986155080440993051970712034649156233308725994139181776241470
discriminant (Δ): 27208233838008087531724099910146002763334065506375895709437142576736624766565380575674285535615824722900873405869848528865141286156987645030029496711594764360268032000000
Faltings height: 31.3725
naive height: 402.6679
primes of bad reduction: 2, 3, 5, 7, 17, 19, 41, 47, 59, 79, 90092142131657, 571015665669101, 61929360015089325767196318530102818683472407988327487857313931709443315397458952478904261605957483653
regulator: 580399477327359370660081778449056723.258714795194993212842046437079354967733454984200980697072000
submitted by: 7fff-zip
submitted at: 2026-09-21 23:10:34 UTC
last updated: 2026-09-21 23:10:34 UTC

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

(124564300753411946688457687724976/10201, 26293975832137872398986044967522147813351494552/1030301)
(12330447154005447734906595450, 76456530057893572369260838855441230152370)
(2917076464667683974904845082363320/231361, -16517658451487597750625770116617444053237362087920/111284641)
(670361894946243079917850269390, 548616558477702695580338369730412364272369230)
(-767478550493862285386037825/4, 14406976834772627181374230467970534433412585/8)
(-6738273118991437196290990281240/289, -438897609686191012536797058803461459016575560/4913)
(1489857017094980558131979268840/121, 94258976283850256848326785861338648716557280/1331)
(12680270909344417068532635090, 164824894515618851367860418619646641069530)
(24580346787559311506240459506368/2209, -605322882198592337157671812022182977206092872/103823)
(12338443334785807518613058220, -78935914496714752054687268221056632605620)
(6579038828947862940650577780, 873959425138644863852174973230004178779780)
(823217170934843676528729715000/81, 186761791224806164525033367107024402145209720/729)
(19112272411870868431022372940, 1530970882197092245051181076993823613067180)
(1311104374030494290806383938746110/1369, -47463527523999485242154258859204784991878099529610/50653)
(77564991964081671015628114212, 20932192125913047474370920424450225962235476)
(127877674700927676104725172963370/10201, -135310133187823420873259592698589875049787669500/1030301)
(-17857088749395138325583004709060/5041, 764831695360093211704844881126769236463892646180/357911)
(2057952071841787859347768631628918568028336633931125550/22025780903523438734942209, 2888254537986156741744809795936154416977438357583310875165717916201693290306414610/103370584579369324211649816715710844927)
(704170287204636989400733864972693305840/11964203161, -17704194353900767103518894626654420978540912715290272590920/1308656505953341)
(8846444809073192625757600775352927328948151088623580/795006334058670014503369, 3780237822367797691266202085353240718923185008538483119494054840081229054100/708852512830405303593076631735588347)
(1651271644969165060288530270795366443693779002/129256571096244601, -8637061321259302360673711231708500605382705850276020206050135642436/46470671304007247715698099)
(718357663282467568316911162867580467491780/35166928089241, -381663556806036989454606760661749701409972954042687917864875460/208545897113902390211)
(122713855866916549796866713711840749591325536/12003933736876225, 321819642413877718982933654419138132259698616634038277806734535464/1315180569908820610894625)
(718091465181713948307747344871598728144/29259813025, -14092283655851265286907722954454395072555061737495756215288/5005037316991375)
(3034574150200538974011755063829360551494976849700/135058190122266700129, 3625490657632624369380501089255653603435951900400898662010284800511576340/1569572526676786685753096649617)
(4472440521320152264593348907103463620768385/14427781785664, 9438773204129280404392949533084748121663544421583983603462627565/54802370912411852288)
(249073266243178692962640285159030/11449, -2621690306748177267639029581199078527002473496660/1225043)
(105484354348183899370643950347596406760/10526144409, -305860726727736888944985440164878982348150936808729236960/1079950837930173)
(1303185076359128059166742316230021330273175538605018782453915/42059696780163507905109111169444, 1227819735529212049541467424042792903526371700955647797298358304404490708159416521190511145/272771634522851054996700041200395487328904630872)

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