curve #1984

y² + xy + y = x³ − x² − 77716552139484187797942108226401075330x + 262884720449903748934282837641219029955148878343472775297

a-invariants

[1, -1, 1, -77716552139484187797942108226401075330, 262884720449903748934282837641219029955148878343472775297]

rank (lower bound)

≥ 24

torsion subgroup

trivial

conductor (N)

1661437540594472610170289527190042676788121744515052516330239539754449278718668745727350050

discriminant (Δ)

186607833959249791149151624223480399920806591732830448743988446899474164648103015342472295275012266652398336000000

Faltings height

20.5960

naive height

273.3520

primes of bad reduction

2, 5, 7, 11, 19, 127, 137, 223, 227, 307, 337, 461, 593, 1319, 522283, 363990268427, 635852046329, 5718659700887713323405615272213549

regulator

10070069179426259601301122.041050573966411878950078717422977677792381946032473127235616

submitted by

Ritik Jain

submitted at

2026-09-25 23:06:03 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-25 23:17:16 UTC

Witness: 24 independent points log in to add more points →

(234924095309746994575561/59049, 58574759678121378797924654058361685/14348907)
(15523172430716483116819/3721, 762099161243785692303029896658355/226981)
(4555952018379429739, 1838031997695471955707036055)
(32498574735921376641493939/8208225, -97668012064451403861079209040646672267/23516564625)
(2699593590567990015/4, 116135466265657241310043628743/8)
(5265020486904702689009/961, -37070544481732502402498153003135/29791)
(-437572974448625936028139/97969, 699858216190290908678375796138282955/30664297)
(43705138586800654441/9, -479345452812933726235703675/27)
(676012925088992317921/289, 47606259163418433381758367704015/4913)
(3082860072992082083, 7252244330010987227590261045)
(5482355863908803489, -1262478108865086545533764545)
(426697127887966776065, 8812266157108379451770344026623)
(3440071116826283802235/324, -146921816663904041214035336565139/5832)
(459838977602146331473/81, -1571758627248895931433263017945/729)
(-9309000242027914511, 13403490952028890945823543455)
(4213063250475206907/4, 107984293168680638407680685585/8)
(10941740004972680029102088304014350219118361/1591776627295787888166649, -14684487818404739030430607086291241627783621978618039823666648685/2008275015045721564941950523680810893)
(25184158344506013582744718990091525131189836868030219759786088725643/7812690979443329624648893117383011019155257302361, 147884449334753028552867013561337067199018928584328653287376509804832456152968695935700980989277462239/21837402053322731539331668212770257378219676480247208178760971959001613859)
(-14460107794708692904419276376653437166345571925175728111371633499/2890174168082931266810614135466773506072861649, 3565129549383674947613841352822273633434688745364960216264672239529898227665102291889106051100875/155376746235535296868451450110806204921926490652878624449431343672857)
(5992152736747541228444396500504512231492642342112014184607029682448676330793945842329142156363501468989/626232632896237739930407572248312032765017478275281968583156186218321449069466418881, 9853260793014211098949497607185026480552864159496772863639694823388971856473186980304397101974913060008401500519777292183745636930244197517239563299661515/495568327673241028161718589224466822314842847731666635887502738244479545456863441934983321149650476192067652500237554894066721)
(4395655030859740849736501691783302327973683161/483281394231471319920587449, 186593755798531595556647420438506990929671506369002669815184685235565/10624294813992624061307287632760341673043)
(467029334245058987589311520139393913394582796495054239449333315226377/180003905609376712983697599954742644992240426040889, 21425959522903836111408132494174674088544872265965149994122791228959638582254067784797955354994477134173/2415032014998666588735161829025415014872774778264175761661218099267723538963)
(755028691293680412991605001181/164008800400, 108606801143242193759363102543150520894049081/66420283985992000)
(-212295802766212246191406588741470183029054912530486644024354433600165079/63446961574781356094534036438812075038409974630665329, 352123664684759140879288430337861043268925877268299762248041355981211238788534765511487793405068073933103825/15981452317334524339040075128595270360511315289094726365757080222774523175627767)

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