curve #4988

y² = x³ − 2419628240966605900x

a-invariants: [0, 0, 0, -2419628240966605900, 0]
c-invariants: [116142155566397083200, 0]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 468368065958652117289244717273184800 ☆ record for quartic twists, rank ≥ 11
discriminant (Δ): 906621279648371737004856847043431749663075471576256000000
Faltings height: 9.6186
naive height: 138.6040
primes of bad reduction: 2, 5, 7, 11, 19, 23, 47, 73, 229, 613, 1493
regulator: 1057130905635.42882907245550715514854717522180497359861784009
submitted by: The Wren
submitted at: 2026-10-21 01:17:03 UTC
last updated: 2026-10-21 01:17:03 UTC

Witness: 11 independent points log in to add more points →

(-1300645170, 30770199045300)
(-1211146650, 33969433808100)
(24163875670, 3748419338819100)
(-128172384979100/120409, 1546196246464613366700/41781923)
(-6939299057130/17161, 67901795768726181900/2248091)
(-2284181011530/72361, 170080568064360059700/19465109)
(-379917716570/14641, 14035548817692719100/1771561)
(-5134288775/64, 7123871883911475/512)
(6264204545/4, 57413121892575/8)
(42746199430/9, 8350427448441100/27)
(1739174135658470/1062961, 22489792679397797634300/1095912791)

Commentary

Rational 2-isogeny partner of David Renshaw's curve #4986: https://elliptic-rank.icarm.cloud/curve/4986. His source is the Aguirre–Castaneda–Peral parametrization at tuple (89,102,168,214), from "High rank elliptic curves with torsion group Z/(2Z)", Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3. Points transported by the rational 2-isogeny and reduced with PARI/GP. Eleven rational points checked exactly; PARI/GP returned rank bounds [11,11]. Rational torsion C2.

last edited by The Wren at 2026-10-21 01:17:03 UTC · history