curve #5970

y² + xy + y = x³ − 47745436418529x + 126983092945543320452

a-invariants: [1, 0, 1, -47745436418529, 126983092945543320452]
c-invariants: [2291780948089369, -109713395742620851004797]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/6ℤ
conductor (N): 33127795610
discriminant (Δ): 109618302873088494976400 ☆ record for ℤ/6ℤ torsion, rank ≥ 4
Faltings height: 5.9374
naive height: 106.1043
primes of bad reduction: 2, 5, 7, 13, 19, 277, 6917
regulator: 5499.680356228010328979227066052181515662194034
submitted by: Lewis Combes
submitted at: 2026-10-27 07:19:08 UTC
last updated: 2026-10-27 07:19:08 UTC

Witness: 4 independent points log in to add more points →

(319505392567/80089, -45192584922242/22665187)
(55057591, 405430756254)
(3661308426359/917764, -1753759063162297/879217912)
(145582140976/36481, 16233077472629/6967871)

Commentary

Found by a rational-parameter search of the Kubert-Tate family for torsion 6, at t=1/6916. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by c=t; b=t*(t+1); converted to the global minimal model. Torsion Z/6, rank exactly 4 (GRH): the four points are independent by the mod-2 Kummer-map test and the 2-Selmer rank is 4. Conductor 33127795610 = 2 * 5 * 7 * 13 * 19 * 277 * 6917, minimal discriminant 2^4 * 5^2 * 7^2 * 13^2 * 19^2 * 277 * 6917^3 = 109618302873088494976400.

last edited by Lewis Combes at 2026-10-27 07:19:08 UTC · history