curve #3582

y² + xy + y = x³ − x² − 4370576197248492759537220356924166830230x − 59611215831321171068555321265886250619474234373793543925228

a-invariants: [1, -1, 1, -4370576197248492759537220356924166830230, -59611215831321171068555321265886250619474234373793543925228]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 519230428950677293578730527990684406193342087025
discriminant (Δ): 3808023231325566756786691244066517471249300112623658597991978332128576065559731252113668556287333611405080041563797265625 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 12
Faltings height: 21.8347 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 12
naive height: 285.4407 ☆ record for ℤ/2ℤ × ℤ/2ℤ torsion, rank ≥ 12
primes of bad reduction: 3, 5, 7, 13, 19, 31, 37, 43, 47, 53, 79, 109, 113, 137, 167, 179, 263, 337, 823, 1171, 1471, 3461, 6269
regulator: 46474535308944672.175425713131857189068791080687541855985334274
submitted by: Matej Jurasić
submitted at: 2026-09-28 21:02:27 UTC
last updated: 2026-09-28 21:02:27 UTC

Witness: 12 independent points log in to add more points →

(-400438207751884348564580515/25684624, -8960990825119638971794189623041990377923/130169674432)
(10303912351188598634891/4, -1045585145936016434232217374727395/8)
(412704018312123506379, -8272259911532268421961425218190)
(197951657429688072893381459/1999396, -1954913971570042585175222857906851754485/2827145944)
(-14086329290291168398306944093858281/244796345948961, 140640008884047141467397761196911163057400599247540/3830076040030719388209)
(-2369442066995301837162024853025/41981551236, 733704347709681193042436039774771555582660297/8601767958948984)
(6944718771074236772885890021/95648400, -71359439788984114913582995610867979553721/935441352000)
(45238516981088592060651/625, -891302751775308498569492246112474/15625)
(5078562337614607619727592559/1452025, 361853921198510509651214366869735640106192/1749690125)
(-5775134324264871998796387239689/328877016484, -20419764708047755793331227773490059191830449755/188603733659211352)
(-30976141141425795358834442483941/553961649796, -41105845566151974971328462986096864786794106265/412305900480065656)
(87074803433113309987476353799/274200481, 25106987041393043679289385969561557815072240/4540485764879)

Commentary

2-isogenous to curve#3562, the fibre at t = 83497/251378 of the Z/4 fibration E1 of X163 from Elkies–Klagsbrun (arXiv:2003.00077, Sec. 9), who list this t among their rank-12 parameters.

last edited by Matej Jurasić at 2026-09-28 21:02:27 UTC · history