curve #4763

y² = x³ − 1033477836241777x

a-invariants: [0, 0, 0, -1033477836241777, 0]
c-invariants: [49606936139605296, 0]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 34178446016095527602163203767328 ☆ record for quartic twists, rank ≥ 10
discriminant (Δ): 70645332869641578355990850250706803718922523712 ☆ record for quartic twists, rank ≥ 10
Faltings height: 7.6790 ☆ record for quartic twists, rank ≥ 10
naive height: 115.3287 ☆ record for quartic twists, rank ≥ 10
primes of bad reduction: 2, 73, 673, 2129, 2393, 4129
regulator: 3788061946.309255772494655922282703222398654213079489567999
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:02 UTC
last updated: 2026-10-20 21:29:02 UTC

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

(-25767824, 97576307532)
(77037849, 614483231376)
(-1001189475652/32041, 242248306123495746/5735339)
(-1722965636/81, -81043257689506/729)
(-41069874752/1681, -7117927801690344/68921)
(-2883562839121/273529, -14106335317318398144/143055667)
(18223731169/400, -1743206625311697/8000)
(14157230633449/329476, 35344219980706467501/189119224)
(-26598836324/2209, 10738734889670358/103823)
(-25009353/49, -7876679054448/343)

Commentary

Quartic twist y^2 = x^3 + a x with a = -1033477836241777. Source: Aguirre, Lozano-Robledo, Peral, 'Elliptic curves of maximal rank', Table 2 (Rev. Mat. Iberoamericana proc. 2008; preprint https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2014/01/ALP-2-23-07.pdf).

last edited by David Renshaw at 2026-10-20 21:29:02 UTC · history