curve #4917

y² = x³ + 6103963510695711647941236360x

a-invariants: [0, 0, 0, 6103963510695711647941236360, 0]
c-invariants: [-292990248513394159101179345280, 0]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 149033482159618868502058245714126232025398782901544198400
discriminant (Δ): -14555118991587742373051858942360454073739663548506909087714094641261415431698141184000
Faltings height: 15.0307
naive height: 203.5498
primes of bad reduction: 2, 3, 5, 13, 23, 31, 67, 89, 307, 317, 337, 1223, 2153, 10657
regulator: 2143188679959624727.208417489912078208017205811128407134080090183
submitted by: The Wren
submitted at: 2026-10-20 22:51:16 UTC
last updated: 2026-10-20 22:51:16 UTC

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

(70666199129761/25, 16429936844221620877741/125)
(3852288531175249/4, 239884719025062710481817/8)
(1850760618334010161/76729, 8535137139960794827843781371/21253933)
(525361621651445569/121104, 6868502824552041304132600607/42144192)
(28181810869748909569/49284, -150997434338290516124340380993/10941048)
(925246452299387761/3721, -932889411841948625690287771/226981)
(48050866213379450521/470596, -419402569274684175696610648189/322828856)
(6462471871029001/324, 2100202778453363782333019/5832)
(421903082123050603391449/11873589156, 661942790166104232875791500365090083/1293817515972696)
(2256475331470677166201/4137156, 108282036692345010576664681209331/8414975304)
(201697482782990041/225, 90927217426347269629029161/3375)
(24900019762354801/400, -6305275226808462194440951/8000)
(2080554156623651739772222009/25797297442816, -132127646679682537253682759085210614444013/131027156630996516864)

Commentary

Obtained by a rational 2-isogeny from curve #4772, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4772. The source commentary attributes the original curve y^2 = x^3 - 1525990877673927911985309090*x to Aguirre, Castañeda and Peral, High rank elliptic curves with torsion group Z/(2Z), Mathematics of Computation 73 (2004), 323–331: https://doi.org/10.1090/S0025-5718-03-01547-3. Applied the standard degree-2 isogeny with kernel generated by (0,0), sending y^2 = x^3 + A*x to y^2 = x^3 - 4*A*x. Here A = -1525990877673927911985309090, giving y^2 = x^3 + 6103963510695711647941236360*x. The coordinate map is X = (x^2+A)/x, Y = y*(x^2-A)/x^2. Thirteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion C2. Same rank and conductor as the source; conductor 149033482159618868502058245714126232025398782901544198400. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:51:16 UTC · history