curve #3084

y² + xy = x³ − 21535474397096264339065986304691195x + 1216367428572439732435342327387827873034132780259025

a-invariants: [1, 0, 0, -21535474397096264339065986304691195, 1216367428572439732435342327387827873034132780259025]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 20468006001635161782527819002994903297490
discriminant (Δ): 44141760873677415861571794333222999999249119945589980584303467843847382090800962970497125039062440000
Faltings height: 18.3950
naive height: 248.7786
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 59, 151, 173, 271, 4273, 697441, 1256873
regulator: 6096263023.946316026307705651387466008600365798475758355355
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:36:29 UTC
last updated: 2026-09-26 20:36:29 UTC

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

(98693634389304010, 7229935746013386672926905)
(87661719875151400, -1474432969930603755102305)
(86718259817155120, -987312232663747713637145)
(90950437786522930, -3169700445461425815605435)
(86003734012146370, -612430086349329528055895)
(19375956111356666347460562460/70527956041, -2371795182192450062322259941196788672145615/18730179813764411)
(793645949064526360/9, -47051199638300693554094365/27)
(342556563872039923/4, -3305367983268564516426529/8)
(140483205699807231902740/1646089, -496193825395397786712406444867015/2111932187)
(6871825747499521202904576161175859735/80526425155964351524, -166201269419155881394315588772288222942964809805106865/722616093445217921000098518232)

Commentary

2-isogenous image of D-z4old9101112-024, rational kernel x=-42159575397842210. Source: https://dujella.github.io/tors/z4old9101112.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:36:30 UTC · history