curve #6221

y² + xy = x³ − 81845018367208852218685574625x + 9015937702651381495267247560797168531015625

a-invariants: [1, 0, 0, -81845018367208852218685574625, 9015937702651381495267247560797168531015625]
c-invariants: [3928560881626024906496907582001, -7789770175090799504752224331566113356158873001]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 75862104980611350173204985583030
discriminant (Δ): -28193914363365699641381441974506028786600218840668850270598542356510308437520384000000
Faltings height: 15.3527
naive height: 211.3383
primes of bad reduction: 2, 5, 7, 11, 13, 17, 19, 79, 101, 181, 281, 283, 463, 30493, 14470627
regulator: 1630515037.177271660016226289856735400325534506221000814008
submitted by: apkorte
submitted at: 2026-10-31 00:12:32 UTC
last updated: 2026-10-31 00:12:32 UTC

Witness: 10 independent points log in to add more points →

(-201497827396950, 4162504436695081932875)
(-54899762927100, 3652908841311282882875)
(159512558854250, 138952592307346300875)
(168434466217310, 94498923595041737215)
(181069167319950, 364513211593849647875)
(219600571238650, 1277823467577151578475)
(1018209490148760, 31325864655706621970515)
(263719807617341530/1369, 31960500852372138058496575/50653)
(367283523676381650/1849, 61386101371066007920211425/79507)
(3028093554567630458/11449, 2968183131492118982704222961/1225043)

Commentary

Specialization at T = -1445/1111 of the fibration given in curve#6190. Found by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-31 00:12:32 UTC · history