curve #3334

y² = x³ + x² − 2386260462382025228095x + 44866777866957132073671901969850

a-invariants: [0, 1, 0, -2386260462382025228095, 44866777866957132073671901969850]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 1185557421935264818440
discriminant (Δ): 1339963598635962003158421351517155890816106884858381250000
Faltings height: 10.6933
naive height: 159.2856
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 23, 31, 67, 89, 193, 347, 2039
regulator: 7534307.037734179442494821425851534602589587876112350
submitted by: David Renshaw
submitted at: 2026-09-26 21:01:32 UTC
last updated: 2026-09-26 21:01:32 UTC

Witness: 7 independent points log in to add more points →

(708096349979/25, -4390220390578083/125)
(12478492695610/441, -249868351726662500/9261)
(30449171435, -661917974787225)
(53672162954, -8450116866863100)
(130631164792874/3481, -586887323235576456600/205379)
(27136418855435/961, -298924064887865625/29791)
(658848929570435/22801, -696275010485001128025/3442951)

Commentary

Rationally isogenous to curve#1830 (isogeny class computed with PARI/GP ellisomat; global minimal model). The witnesses are the images of that curve's witnesses under the isogeny, so they remain independent and the rank bound 7 carries over. Torsion ℤ/4ℤ.

last edited by David Renshaw at 2026-09-26 21:01:32 UTC · history