curve #4773

y² = x³ + 29084092124144031825559131423x

a-invariants: [0, 0, 0, 29084092124144031825559131423, 0]
c-invariants: [-1396036421958913527626838308304, 0]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1323698042444241813123724539916763571371277542214384672
discriminant (Δ): -1574513935558173533083864989903024250534304858231944822112438363712683994078696754173888
Faltings height: 15.4210
naive height: 208.2336
primes of bad reduction: 2, 3, 7, 11, 13, 19, 31, 37, 59, 79, 223, 599, 919, 1871, 2903
regulator: 7368740603302.4545537957003327274177702390166138522137825731464
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:50 UTC
last updated: 2026-10-20 21:29:50 UTC

Witness: 12 independent points log in to add more points →

(12310556095473727/81, -2052990864449262770385808/729)
(239067343399872, 4540541714194976405448)
(20684427680631, 781305023985244296252)
(489388396494612433/19321, -2330355868192118945932510484/2685619)
(7786014892179601/36, -874918363083623177631697/216)
(81933434051953, -1712596235382928568464)
(442489649227498873/149769, -16992857133628121629124585344/57960603)
(113147149220625, -2176998690911041537800)
(3139601972092663, 176177991751057927251236)
(1410294126554751/529, -3388373954648688170532912/12167)
(268540855122324648721/1585081, 6244098006399203778653546738732/1995616979)
(15264229403426803228/76729, -78550476509928003934462429034/21253933)

Commentary

Quartic twist y^2 = x^3 + a x with a = 29084092124144031825559131423 (2-isogenous model of the curve listed there). 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 21:29:50 UTC · history