curve #5078

y² = x³ + 229187948913x

a-invariants: [0, 0, 0, 229187948913, 0]
c-invariants: [-11001021547824, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 373526157702740607157824
discriminant (Δ): -770469245543204061420401721362207808
Faltings height: 5.5755
naive height: 90.0870
primes of bad reduction: 2, 3, 7, 11, 19, 23, 59, 101, 127
regulator: 27056157.33850264440756526992451194868436448589250606
submitted by: The Wren
submitted at: 2026-10-21 02:16:05 UTC
last updated: 2026-10-21 02:16:05 UTC

Witness: 7 independent points log in to add more points →

(799236, 832892418)
(941017/16, 7486303517/64)
(19442500/9, 87808940450/27)
(1320976273/3136, 72680717760839/175616)
(17849964825/13924, 2545688780812485/1643032)
(27457795057/54756, 6290393640633487/12812904)
(43021617408/104329, 13672927024794576/33698267)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (12,15,31,36), reduced by rational fourth-power scaling. Reference: "High rank elliptic curves with torsion group Z/(2Z)", Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3. Selected in a bounded tuple search using a small-prime sieve and height screening, following the search approach described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Seven rational points checked exactly; PARI/GP returned rank bounds [7,7]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:16:06 UTC · history