curve #1912

y² = x³ − x² − 3969961x + 2864359465

a-invariants: [0, -1, 0, -3969961, 2864359465]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 21524842560
discriminant (Δ): 463318847232787353600 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 5
Faltings height: 2.7145
naive height: 57.1964
primes of bad reduction: 2, 3, 5, 13, 23, 31, 41, 59
regulator: 90.7101828331645530012470143478129818835379076307
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-25 18:24:54 UTC
last updated: 2026-09-25 18:24:54 UTC

Witness: 5 independent points log in to add more points →

(-2272, 12285)
(-2192, 32085)
(-1991, 53592)
(-1752, 66625)
(-1517, 73440)

Commentary

Found on 2026-09-25 the family y^2 = u(u+a)(u+b), with a = 460, b = 3658 and x = u+1373. The five submitted points were verified to be independent modulo torsion by exact full 2-descent: the square-class matrix has rank 7, including two torsion rows of rank 2. An additional certificate using quadratic characters at good primes gives the same ranks. The torsion subgroup is exactly Z/2Z x Z/2Z, verified using good-prime reduction counts. This proves rank >= 5. Comparison data were obtained from the ICARM Elliptic Curve Rank Leaderboard, supported by NSF Grant DMS 2425401.

last edited by Natanael Wildner Fraga at 2026-09-25 18:24:54 UTC · history