curve #6421

y² = x³ − x² − 53603271818126921x − 1423611811217633798513079

a-invariants: [0, -1, 0, -53603271818126921, -1423611811217633798513079]
c-invariants: [2572957047270092224, 1230000620329777885535853568]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 7290134835250249699654080
discriminant (Δ): 8981685139825832320914599277720264174856510649241600
Faltings height: 8.6605
naive height: 127.1748
primes of bad reduction: 2, 3, 5, 11, 13, 47, 103, 179, 263, 401, 409, 883, 1609
regulator: 1243485.34741533757334716008330936625413688944621291723
submitted by: The Wren
submitted at: 2026-11-02 14:17:06 UTC
last updated: 2026-11-02 14:17:06 UTC

Witness: 8 independent points log in to add more points →

(-204092959, 1007553209520)
(-11230321169/81, 1332917308765900/729)
(-195527272, 1257816886599)
(-102669232, 1731346733301)
(1665745661, 67314528824400)
(22313810261/49, 2840928806634360/343)
(-38907633631/784, 23173019356076535/21952)
(-174604619/4, -7301765144635/8)

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=-110/13. 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 [8,8]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 14:17:06 UTC · history