curve #3979

y² + xy + y = x³ + x² − 509021299755831291275340601139x + 111151221092896624541690625755717516434042481

a-invariants: [1, 1, 1, -509021299755831291275340601139, 111151221092896624541690625755717516434042481]
c-invariants: [24433022388279901981216348854673, -96034655024262866851688612752204793321629113785]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 27145429606258964067875985943433158218
discriminant (Δ): 3103697639258640422000292758398416206001431250963213987009342036381344363679941004939300864
Faltings height: 16.0782
naive height: 216.8205
primes of bad reduction: 2, 3, 7, 13, 17, 19, 23, 103, 1063, 6673, 43801, 76441, 1625287, 1683223
regulator: 1802137171296770.69510759650887893800781808223482426049360312
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:38:33 UTC
last updated: 2026-10-05 04:38:33 UTC

Witness: 11 independent points log in to add more points →

(17173919075106975/361, -63992347597495589314680782/6859)
(23032874262803275/9, -3369067144167854551518646/27)
(-508257524050473098721/697225, -5669564725966450137578975474378/582182875)
(6651262448027715219/30625, 17648737777882372740712762978/5359375)
(-9647460810771565773/152881, -714876116876242922877608372318/59776471)
(-7505832468440628262929/14707225, -870141737480050514756955139257086/56402207875)
(2375207556564020128327/3876961, 41285302049229108340528366629878/7633736209)
(803962337910917113069791/4628761225, 1665789358793406700449900685565735506/314917769942875)
(1278520983170868265268299/2134902025, -453044534141004393641874324330855242/98643148065125)
(129201686003041749452880823/523174129, 1468589640166001412891727497450993953990/11966561852617)
(-6651223221016905641584914968349825/8847101826137811721, -218456973748942021813869640192083223242981387205358/26314888776376671145389752731)

Commentary

Unlisted rational-isogeny neighbor of curve#3518 (source contributor: dujella). Source equation and witness provenance are credited to that entry. Exact rank 11: 11 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 04:38:33 UTC · history