curve #4003

y² + xy + y = x³ − 957249998989722230269291478x − 2233397097131443816082419687178453392744

a-invariants: [1, 0, 1, -957249998989722230269291478, -2233397097131443816082419687178453392744]
c-invariants: [45947999951506667052925990921, 1929655091921498535095283349721604342344219]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 4858903795077400145101018027817292162914370
discriminant (Δ): 53983048570875774740525271809583428053599358575990413808988770782244687155894010000
Faltings height: 14.5652
naive height: 197.9919
primes of bad reduction: 2, 3, 5, 11, 19, 37, 67, 103, 229, 271, 631, 643, 1093, 1103, 1193, 1237, 1489, 2293, 19843
regulator: 6487225235337241158270.68216740473885625079302905424341466004412313
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:43:46 UTC
last updated: 2026-10-05 04:43:46 UTC

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

(-1361278550053487880249163995/48134331527881, 15732159160861853979836686676341184013787/333950745498301838971)
(-155059079827794491306055/22468810816, 214178405795602248901830031244025127/3367984866075136)
(-2682947943672907932184210/282108637321, 11616328968521372057862608736600728028/149838899518038619)
(716187499591874921205/22174681, -2427272395298280949432776004387/104420572829)
(-1628435019821927179633410/470970630529, -10398609453203688291984651717149186476/323214427525028417)
(-523420889872127303410114064335131/172294300511989446025, 57509234695918683415332277350502622852905278488443/2261550900760497253309437638875)
(-21233724933258593172403419/9015192405961, 77385447329538595836858882715979509499/27068394669862487291)
(-1519290464871018566161554775/64850631834121, -44725397512972109273749493635838679717387/522241424803226237669)
(2647484481346247029256235642414/31283473389594409, -4001011468146731485860270217099052713830557668/5533150077256218903498923)
(2291205990784683476694030/736525321, -3467965044605438066988238362244898657/19988560686619)
(4033100076491435438875160732093745/3734325466685522944, 256023497050828567993343309196263468202684832061159/7216360980327506363333967872)
(-6519354405981770537864007573210/690100332226591729, -44282521139319985994335898738177027947962467939/573282064395896874360111433)
(21696126292770731331478773182012026030/230196664187134840621881, 95315569175293882895270887423764787751673842003816870788/110445630520624794443240629028530029)
(50480422704150412098198122641510479539145/526109874906179070145300489, 10720887367888138727100156220485692752166683882950133489450411/12067426995857467705252219765357570548763)

Commentary

Unlisted rational-isogeny neighbor of curve#3641 (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 04:43:46 UTC · history