curve #6391

y² = x³ + x² − 210973303610686295544609x + 35969449082416267703539263137015391

a-invariants: [0, 1, 0, -210973303610686295544609, 35969449082416267703539263137015391]
c-invariants: [10126718573312942186141248, -31077604007268415607297801003498145280]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 122783327132041139329765184064
discriminant (Δ): 42061263808872091731395544555978495743799205361567942482466440132952064
Faltings height: 12.3174
naive height: 172.7317
primes of bad reduction: 2, 3, 11, 19, 29, 31, 37, 103, 131, 241, 691, 4133, 24337
regulator: 86353562.90453214861084637467281702531842410834191689552569
submitted by: The Wren
submitted at: 2026-11-02 01:11:52 UTC
last updated: 2026-11-02 01:11:52 UTC

Witness: 10 independent points log in to add more points →

(906841244151, 768375602493668928)
(14361772553985/529, 2116560235865803776528/12167)
(423090947253, 149814388538330340)
(328319262135, 45755267822136576)
(173648642478855/529, 555793032273821415552/12167)
(193656805426718631/237169, -73786294961704598685920640/115501303)
(4848229160679/49, -43472674873269291264/343)
(316556950266, -30100136503907943)
(-39285743975615/81, -113412422032147483696/729)
(5934649309572111/17161, 148579297168106347549248/2248091)

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=-121/592. 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 [10,10]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:11:52 UTC · history