curve #6398

y² = x³ + x² − 429541512962710772361827721x + 2497540864242394806249888229990723409655

a-invariants: [0, 1, 0, -429541512962710772361827721, 2497540864242394806249888229990723409655]
c-invariants: [20617992622210117073367730624, -2157875306705552820555636691414425232325632]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 58255187586444887551054174315287359040
discriminant (Δ): 2377497812967454207170728533165398296865227704660092419002363149181356572991590400
Faltings height: 14.3225
naive height: 195.5879
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 31, 61, 107, 109, 131, 191, 419, 439, 823, 977, 2281, 7057
regulator: 1394043449.59525687426335219684430921566091664953649984383192
submitted by: The Wren
submitted at: 2026-11-02 01:20:09 UTC
last updated: 2026-11-02 01:20:09 UTC

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

(575116627151607, 13783342812814275802200)
(5468058539187, -17671328202295825200)
(-2473075722198, 59537416508258164515)
(6185207051319, 8795670162923335224)
(-11603673713709, 76937773250129223264)
(21887711530617, -59846871588730673460)
(2904563984833695/49, 147657042726988588796640/343)
(-15847431444813, -72970743736745838000)
(5114615798889537/256, -178095136316367588618465/4096)
(76857076697390370/12769, 16462771919588462883813255/1442897)
(-23124207348273, 8072447067621303840)

Commentary

Z/2xZ/2 X400a family y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q), at T=p/q=1045/17. Family equations as posted by wgxli with curve #4252; section formulas from the Layr Labs Elliptic Rank Challenge repository (submission/research/x400a-family.gp, commit 7482bf9). T selected by our Mestre-Nagao sieve; PARI/GP 2.19 ellrank seeded with the family sections, bounds [11,11]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:20:09 UTC · history