curve #4776

y² = x³ − 19348006334886975416600173605900x

a-invariants: [0, 0, 0, -19348006334886975416600173605900, 0]
c-invariants: [928704304074574819996808333083200, 0]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 29947627930786122521100129658764695285207276277281520681184800
discriminant (Δ): 463541515935750417518113098687641441640539948171759286937537434489666615895631124560216256000000
Faltings height: 17.0461
naive height: 227.7340
primes of bad reduction: 2, 3, 5, 7, 11, 23, 41, 61, 89, 101, 643, 1361, 3491, 6091, 87049
regulator: 942031705578444.3433585981304697090534852570755731138135504864006
submitted by: David Renshaw
submitted at: 2026-10-20 21:30:21 UTC
last updated: 2026-10-20 21:30:21 UTC

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

(-6685244291066, 11373031702830608827148)
(-3088886477780425055/2304, -16964060301495901701616519025/110592)
(1113926190754693030/9, 1174925120567678344989109700/27)
(-12355545911121700220/33489, -515962527706177138397991232900/6128487)
(637326309322684368409/2704, -16086706177688691870266035208323/140608)
(507883368159829270/81, 257942570864429461262473900/729)
(50390034811827670, -11268238228933783941637900)
(-143134500408212114780/32761, 199622478750220376880505105300/5929741)
(-572165430599501760132650/150822961, 253987795014603580978122982191338900/1852256784041)
(710104289787079453750/28561, -18624213615344080555467798851500/4826809)
(-3418489052792751770/841, -2613721914191026499926866500/24389)
(-2763470101327732775/5476, 39776970522721584576053409725/405224)
(26439644613419305/4, 3209032577947349933896075/8)

Commentary

Quartic twist y^2 = x^3 + a x with a = -19348006334886975416600173605900. 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 21:30:21 UTC · history