curve #4642

y² + xy = x³ − 5389172695796492879808988664173495088495x + 139775541244578540291788629334314438134569541055324861316025

a-invariants: [1, 0, 0, -5389172695796492879808988664173495088495, 139775541244578540291788629334314438134569541055324861316025]
c-invariants: [258680289398231658230831455880327764247761, -120766067635315858812493396178945022035614330655621171823417241]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 232655514909455128819343175296085744158021265811427070
discriminant (Δ): 1577127184921640023068235036024080067291708302061776211382714398239870955501019243097386499486331320478447601591629440000
Faltings height: 21.8031
naive height: 286.0692
primes of bad reduction: 2, 3, 5, 7, 11, 17, 41, 43, 47, 59, 61, 97, 109, 181, 251, 311, 331, 349, 461, 809, 971, 3347, 4621, 5849, 35141
regulator: 424887355739.29037164814025871614918311428362091223075252575306
submitted by: The Wren
submitted at: 2026-10-19 16:13:00 UTC
last updated: 2026-10-19 16:13:00 UTC

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

(664589994607916890, 369045044035967441916035800705)
(15799930667961736732, 242014971366598387635238147831)
(52832493462030148060, 50219408818337903104175452855)
(12692726059081896760, 270955926124550613629030249635)
(54675210776482909510, 92554293513238852175268721885)
(-50791491800014834874, -531478135746980346031650880163)
(9857969417203597510, 295985193486239079363212648605)
(-29634988970924006528, -522931355023613456936377786859)
(70879298019141692230, 337466278335892123131338586205)
(31944600286439789230, 14787837858166493367196911205)
(-76126424263489277858, -329944938565453669610599577579)
(2213395861675027686730, 104076443455236884054187957598705)

Commentary

From Andrej Dujella and Juan Carlos Peral’s prescribed-torsion rank table, z2z2old1011121314.html: https://dujella.github.io/tors/z2z2old1011121314.html Twelve supplied rational points checked exactly on the displayed model; the source lists them as independent. Rational torsion Z/2Z × Z/2Z.

last edited by The Wren at 2026-10-19 16:13:00 UTC · history