curve #3997

y² + xy + y = x³ + x² − 63703939810549162353275x + 11868055398390437520149466551679881

a-invariants: [1, 1, 1, -63703939810549162353275, 11868055398390437520149466551679881]
c-invariants: [3057789110906359792957201, -10253999864232271435740936799098596345]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 455514655953569565656462935595492873046
discriminant (Δ): -44302059026454397158902707819785942171747935099609754556932002517086208
Faltings height: 12.2472
naive height: 170.4415
primes of bad reduction: 2, 3, 7, 11, 23, 37, 43, 71, 157, 631, 12721, 1299673, 11326561, 20456041
regulator: 4264672898283794080.108075246423029782146790107717266125583802689
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:42:09 UTC
last updated: 2026-10-05 04:42:09 UTC

Witness: 13 independent points log in to add more points →

(606291373753531/2209, 12740702239957423094602/103823)
(-214860421121379989/1283689, 194270228276011449978517342/1454419637)
(-5332394472259903501/16883881, 1527312611801963048419232014/69375867029)
(65951039939182919/588289, 35343219277989024902933458/451217663)
(3471931727793895343569/4913309025, 194730713318824099722481639922282/344398396107375)
(32234168657830431963179/15689316049, 5747433310781240657270333171051966/1965196660349593)
(5891385860308257769/85100625, 69288261196927339332130918528/785053265625)
(334463113032996239318701/225854008081, 190957472251661221086999322989237886/107335084654422521)
(6189305869374636610699/34661875329, 507566802477085969131287527092286/6453243963127233)
(96091505789278499/410881, 26020638055766179764028934/263374721)
(7996251751564752417280199/139860292441, 22611295276972785225150033707994160166/52304812306792739)
(10737146067711000271/19193161, 32707580005246720261694377346/84085238341)
(-1842340144533003037499140489/11927685782463025, 174881654461860423202346149718847697434106/1302669619023198776883625)

Commentary

Unlisted rational-isogeny neighbor of curve#3637 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 13: 13 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 04:42:10 UTC · history