curve #4975

y² + xy + y = x³ + x² − 2945073192488637954x + 1890304635597982672357697295

a-invariants: [1, 1, 1, -2945073192488637954, 1890304635597982672357697295]
c-invariants: [141363513239454621793, -1633223206216883378212960126481]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/4ℤ
conductor (N): 62497082243308795337562
discriminant (Δ): 91168936842587529759075406003445642750890433852469215232
Faltings height: 9.5117
naive height: 139.1936
primes of bad reduction: 2, 3, 29, 37, 61, 71, 73, 89, 149, 877, 1193, 2213
regulator: 91443283.0365208502864437321800582167283467164029757787
submitted by: The Wren
submitted at: 2026-10-21 00:50:39 UTC
last updated: 2026-10-21 00:50:39 UTC

Witness: 8 independent points log in to add more points →

(823465257, 4850039878851)
(1181335929, 7732624988163)
(20777418301/16, 1023394739153153/64)
(19727093781/25, 949627685674039/125)
(29443805815/49, 6302094152742065/343)
(10969721335345/1369, 35557373550209234807/50653)
(469752603598489/410881, 1105760314679097161347/263374721)
(5709615809046625/32041, 431409542366921499194881/5735339)

Commentary

From Michael Rubinstein's elliptic-curves-by-rank repository, families/family_Z4.part2.tsv.gz: https://github.com/mrubinst/elliptic-curves-by-rank. Eight rational points checked exactly; PARI/GP returned rank bounds [8,8]. Rational torsion C4. Source-attributed existing curve.

last edited by The Wren at 2026-10-21 00:50:39 UTC · history