curve #5351

y² = x³ − 7236137601x

a-invariants: [0, 0, 0, -7236137601, 0]
c-invariants: [347334604848, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 186174888464377014048
discriminant (Δ): 24249367994023073887071084339264
Faltings height: 4.7116
naive height: 79.7207
primes of bad reduction: 2, 3, 7, 11, 17, 41, 71, 211
regulator: 14101074.56582729467066805068027326735225795104059372
submitted by: The Wren
submitted at: 2026-10-21 15:38:55 UTC
last updated: 2026-10-21 15:38:55 UTC

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

(5221225, -11928915620)
(249001, -116775980)
(990025/9, -624573440/27)
(94249, 12458060)
(22762441/225, 58781983364/3375)
(1555009/16, 937825055/64)
(10426441/16, 33378899525/64)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (9,13,25,30), followed by a rational 2-isogeny and fourth-power reduction. Reference: "High rank elliptic curves with torsion group Z/(2Z)", Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3. Seven rational points transported through the isogeny and checked exactly. PARI/GP returned rank bounds [7,7] for the source curve; rank ≥7 transfers through the rational isogeny. Rational torsion C2.

last edited by The Wren at 2026-10-21 15:38:55 UTC · history