curve #4005

y² = x³ + 6222454764742850429984118x + 18642056625464702040857421297093076981

a-invariants: [0, 0, 0, 6222454764742850429984118, 18642056625464702040857421297093076981]
c-invariants: [-298677828707656820639237664, -16106736924401502563300812000688418511584]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 85468703986756648905003803597594316557982840
discriminant (Δ): -165550670784052849205508341887404385684252761598401932258426570200004082750000
Faltings height: 13.4988
naive height: 185.1601
primes of bad reduction: 2, 3, 5, 11, 47, 53, 61, 97, 103, 163, 197, 313, 463, 607, 1171, 1217, 3041, 8377, 138637
regulator: 34480631533548318057699.0748374427632772532635473725529471265048607
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:44:07 UTC
last updated: 2026-10-05 04:44:07 UTC

Witness: 14 independent points log in to add more points →

(-996403803097108612845155960292762399884863/2083449050919399731275249935376, 11861359190719957958838599442374650770049170667432156241134469025/3007283191431233401107676153337294425337026624)
(-448227613902017373070993119108569266/239426738524654637926225, 76998529475747488809673637701692282090320316819363273/117154499876649637697550924286238375)
(1105506566249104912341601658438/485209824805021849, 2258344105756874867057085248073140514240134325/337982628528456535973869507)
(-1201012240579028200560728939/908762927033025, 78017583733129250633147339359470789511316/27395290025644826577375)
(3055978807482420306327708966/1600325696573041, 391974867099160292138808802690896463392655/64019542788832388849911)
(3141611395537528368328673670549/159244898611770049, 5619416656464539239793307902304951995663037620/63547474130234187186085343)
(29134751651432546795824495077/5198772092505121, 5678880559011288126232040600388725018911600/374844521986343504458831)
(48563863159735605259910908622/162162167272681, 10702495549968037752801155921008218829077525/2065020227240996204171)
(-116425273892684814667480059375627/86763347837533213729, 2268091879153244071452463363201373934935122512700/808173213612989535770100271183)
(22547622018483543945913532656963337/9072737641973252229376, 6077351294337817567687328178400252905538039385242725/864186601655194601648805634789376)
(-26629019776108910225027760038269866/20170109599505056992025, 8165792369926082177002362953723158674208419863470277/2864589443264967978126233014416125)
(3744780377618705380258159783809340517/4828577956793003756641, 7246725047981304760359392063215879733511564105205512600/335528079581406984825059624163311)
(79492058121225775673035481359171451177/15176736053241351251959696, 825474155651893136468453038652475169703442227632090000725/59124512343557753780172682634333160256)
(-2640423753519537367429818412005703/1605044320272403219456, 127857430742698441209976007901766322263287926267475/64302897639587590255281442095104)

Commentary

Unlisted rational-isogeny neighbor of curve#3644 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 14: 14 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:44:07 UTC · history