curve #5244

y² = x³ − 27843101517x

a-invariants: [0, 0, 0, -27843101517, 0]
c-invariants: [1336468872816, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 205021699725214598688
discriminant (Δ): 1381442479670163940363056560346432
Faltings height: 5.0485
naive height: 83.7632
primes of bad reduction: 2, 3, 7, 11, 17, 41, 79, 199
regulator: 73703.49747887285088748790095459605135143626251050450
submitted by: The Wren
submitted at: 2026-10-21 10:59:01 UTC
last updated: 2026-10-21 10:59:01 UTC

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

(-162503, 15275282)
(-3927, 10453674)
(518793, 353816826)
(-39947061/529, 497500697694/12167)
(-391919/4, 338226097/8)
(-269247/4, 316903719/8)
(2939827/9, 4333304998/27)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (5,8,11,18), 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 following the parametrization search 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 10:59:01 UTC · history