curve #4772

y² = x³ − 1525990877673927911985309090x

a-invariants: [0, 0, 0, -1525990877673927911985309090, 0]
c-invariants: [73247562128348539775294836320, 0]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 149033482159618868502058245714126232025398782901544198400
discriminant (Δ): 227423734243558474578935295974382094902182242945420454495532728769709616120283456000
Faltings height: 14.6842
naive height: 199.3909
primes of bad reduction: 2, 3, 5, 13, 23, 31, 67, 89, 307, 317, 337, 1223, 2153, 10657
regulator: 261619711909133.8778330587756240329843771003187412606364843860086
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:39 UTC
last updated: 2026-10-20 21:29:39 UTC

Witness: 13 independent points log in to add more points →

(40502815695250, 68095917826495656950)
(-6327619304591/4, 392735126147523810937/8)
(-2211553703101972190/76729, 3008656553586590275897391890/21253933)
(-5321527835161031/144, 133004860544159607469843/1728)
(-3636467276537160/1369, -3217457963157928795943820/50653)
(947544741895153935/3721, -911440746861495549168800265/226981)
(115337107736760, -1165455112761905953260)
(50289827997240, 224598565448552943180)
(264325378043686609/4356, 103991570672793643083320713/287496)
(177617015126165689/324, 74665747783962957864740003/5832)
(-382291046344334/225, 171689748953423691457286/3375)
(-301160451110375/16, -9504467293840947827075/64)
(-58860725050023936/3721, -32244663522690200213254272/226981)

Commentary

Quartic twist y^2 = x^3 + a x with a = -1525990877673927911985309090. Source: Aguirre, Castaneda, Peral, 'High rank elliptic curves with torsion group Z/(2Z)', Math. Comp. 73 (2004) 323-331, doi:10.1090/S0025-5718-03-01547-3, https://www.ams.org/mcom/2004-73-245/S0025-5718-03-01547-3/.

last edited by David Renshaw at 2026-10-20 21:29:39 UTC · history