curve #4984

y² + xy + y = x³ + x² − 15493463601932144343063x + 1061800377674404890767253227979156

a-invariants: [1, 1, 1, -15493463601932144343063, 1061800377674404890767253227979156]
c-invariants: [743686252892742928467025, -917395526316263472519602360937493625]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 26320902315447160074455486175
discriminant (Δ): -249018842002506028588932237404394542966764817290109391773359367109375
Faltings height: 11.8261
naive height: 165.6137
primes of bad reduction: 3, 5, 19, 29, 71, 149, 571, 1861, 8521, 9391, 708041
regulator: 1181080937.3952202627924101672664919978227291737371126937
submitted by: The Wren
submitted at: 2026-10-21 01:10:03 UTC
last updated: 2026-10-21 01:10:03 UTC

Witness: 9 independent points log in to add more points →

(51220296369, 20064818985008867)
(57086568220, 19062231904403552)
(72047953045, 17875150748233577)
(-1349090887154/9, 108233434670013209/27)
(2611319813245/16, 3434867939092017953/64)
(444838386232534/4761, 6810925680101793860278/328509)
(2473029808864185/4489, 120202609338710037191921/300763)
(7945428485420990/44521, 592696280121796819526802/9393931)
(801615885689940970/2819041, 663472062201634101516379678/4733169839)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, families/family_Z4.part2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Nine rational points checked exactly; PARI/GP returned rank bounds [9,9]. Rational torsion C4. Source-attributed existing curve.

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