curve #1681

y² + xy + y = x³ − x² − 30620874157130056907398299577780504257343397787x + 1813448619568251003382059938724347506907975886406614181834028734506699

a-invariants

[1, -1, 1, -30620874157130056907398299577780504257343397787, 1813448619568251003382059938724347506907975886406614181834028734506699]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

658029205731169758828630675748582184078453585210743780059080274615531705990968043323369990437510625072493661277890

discriminant (Δ)

416849336941821009263883619967198204128729937607841916024631018340432733024871575408621894513902094125217367802428645592132686967367680000000

Faltings height

25.7106

naive height

332.7276

primes of bad reduction

2, 5, 11, 13, 17, 29, 31, 41, 67, 89, 866081, 625576067, 227307212633059886908964359333207525762127846618173168850566814675005192123145332610441

regulator

1663552710230779002742942598833.7448725331888239524420516450790795008183392935641490556121246

submitted by

Ritik Jain

submitted at

2026-09-25 04:08:44 UTC

later contributions

primes of bad reduction recorded · Daniel Błażewicz · 2026-09-25 05:06:16 UTC

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

(501238860651482832265553839/2025, -8834703369379571514479476796953777609802/91125)
(-4586017376582462052183442774515521367597752165210031333/34495043532984810929002908641521, -12044909483496464244327519269710352594692566649789891040754419035720823192790843594/202597949737708984482608051396180833105862469431)
(33703262788248914164557, 28630562720070921151995178321529846)
(-15028650777981652349653, -47647084565101054944410547937601254)
(-166605351389757691697201633/3481, 11562251322319972884617655645845232370334/205379)
(161960596292198779150906483/3969, 6285439017560610136618192657083269617262/250047)
(1114725075863178429657287/121, 52098620241456717954840104523434700926/1331)
(1113065830594668537189557, 1160484794411894336584338740779680596)
(24960629411313630171257427/361, 35498180139541175987633791388157244814/6859)
(3904931648154241967295763/9, -7154086507629767402418321328695874658/27)
(342342427940895093390347, -177348774412926287489039735972289094)
(14014903698650928067948966683/97969, -582254881788879557678304558284572764333238/30664297)
(40470456095206800558758029/225, 155578287090728808339207528212576369962/3375)
(28439320234108799151182263/9, 151434350100795119792288199428114369842/27)
(70804288262443993938947, -564642676697647605085928453757694)
(137038116419307443165307, 13810196246396106119515639274950346)
(78601695686657978552737012134403/1370998729, -796762559012894045163981068050238282877087458882/50763969938683)
(-670504379767629433380149373/6889, 35578561181305300280267437727920358052606/571787)
(53480970244623767870147612830248825593707/265570905338589841, 8451882106150110485206888722414673751643636347968997324816186/136858173979284944835010361)
(-10349240339746456923811891082738894235871/59886656763990016, 646188301711937788391958696728876612380793912164928914536017/14655313166188353796440064)
(7457937744712336017424773718716522171/108680270550289, 6732512783081655677000831554372476693577042989913933334/1132989972922163470087)
(-5170356463263168700219495145453/26925721, 3459054134850966386504624333947160951804661074/139717566269)
(-3180707551389677949629684588962390773046630212970981/40991189508517950941513160121, 506337465890650647191238616088903942144327786598674955969791669557355746642418/8299191420772123232106962110394782727297581)
(-2703387101700370043764744512144347151287323817425013/15178664939912207740000629441, 75208987402234639319970755628403008717238607356910582025086999235171007092794/1870037697445409979044410004599182448083489)
(-97418538230703658558049571181/2493241, 213833407440913140436735827355566478589308142/3936827539)
(-23416823902200896826217213/361, 407333486369840891126007260574634540414/6859)
(-1301707270858153869685448240395620141417/20355945037244644, 172065978474332773522933439742364206777034077502368179012513/2904269435501442096152728)

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