curve #1377

y² + xy = x³ − 305209353709156400979315990x + 2052319031551796164066894280222290328100

a-invariants: [1, 0, 0, -305209353709156400979315990, 2052319031551796164066894280222290328100]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/4ℤ
conductor (N): 3985472961137327010
discriminant (Δ): 96010560714044258693986173065625600000000000000000000000000000000
Faltings height: 13.3920
naive height: 194.5627
primes of bad reduction: 2, 3, 5, 13, 17, 29, 31, 43, 173, 257, 367, 953
regulator: 78354045.9062285649286047818538931300307983312688
submitted by: jesper-petersen
submitted at: 2026-09-24 18:54:35 UTC
last updated: 2026-09-24 18:54:35 UTC

Witness: 5 independent points log in to add more points →

(7899305616504722267724/783160225, -111134593729620941915095302/21916738896625)
(18096668989437231425435435436/1794156535525225, -566697509437646847022180756522013214/75995958541161685059125)
(584407629884600769978739949267020/57939878506167457569, -2479226236479772715276940765046029462634630/441028210669438677673584616497)
(1419002035305293792238082716/140683676830225, 208142510101249240089202413401192694/1668651201138451178375)
(4991467215289781887682928097964/472980963875293225, 841613326852952451025605341333772938900552158/325286090564549507047320875)

Commentary

Obtained from curve #1093, due to Dujella–Lecacheux, via two successive rational 2-isogenies. The first quotient has torsion \(\mathbf Z/2\mathbf Z\times\mathbf Z/4\mathbf Z\); quotienting by the appropriate rational 2-torsion subgroup gives a curve with torsion \(\mathbf Z/4\mathbf Z\). Its global minimal model has ainvs \([1,0,0,-305209353709156400979315990,2052319031551796164066894280222290328100]\) and conductor \(3985472961137327010\). Mapping the five independent points from #1093 through the two isogenies gives five independent points on this curve. The construction and invariants were verified in Magma.

last edited by jesper-petersen at 2026-09-24 18:54:35 UTC · history