curve #4983

y² = x³ − 1009445761048159101650700x + 388764686142554784001936881006134000

a-invariants: [0, 0, 0, -1009445761048159101650700, 388764686142554784001936881006134000]
c-invariants: [48453396530311636879233600, -335892688827167333377673465189299776000]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/4ℤ
conductor (N): 20238386706299310289552929600
discriminant (Δ): 539162885155027115777493333246787081864855310068919051247276840960000000
Faltings height: 12.6134
naive height: 177.4279
primes of bad reduction: 2, 3, 5, 7, 37, 53, 79, 151, 443, 467, 1877, 1951, 11329
regulator: 378536112.29822856088822205981677401486085084735967823798
submitted by: The Wren
submitted at: 2026-10-21 01:08:57 UTC
last updated: 2026-10-21 01:08:57 UTC

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

(-1152867950114, 142268615798523984)
(393737619949, 228798574265141757)
(667120273630, 110654737856506800)
(1193694589480, 940583993162883300)
(9660194212180, 29868384435036877800)
(-6620677546154930/6889, 392518041177057828051600/571787)
(-636391368432770/1681, 58345644581485452298800/68921)
(1910033547214/9, 11584627644937675888/27)
(1629146799219247630/3200521, 472957350474986870947405200/5725732069)

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 01:08:57 UTC · history