Properties

 Label 2.491.24t22.1c1 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $491$ Root number not computed Frobenius-Schur indicator 0

Related objects

Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $491$ Artin number field: Splitting field of $f=x^{8} - 2 x^{7} - x^{5} + 7 x^{3} + x - 9$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.491.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots: \begin{aligned} r_{ 1 } &= -65501842 a - 21851162 +O\left(47^{ 5 }\right) \\ r_{ 2 } &= 34869023 a - 20785166 +O\left(47^{ 5 }\right) \\ r_{ 3 } &= 65501842 a - 55753363 +O\left(47^{ 5 }\right) \\ r_{ 4 } &= -34869023 a + 15523848 +O\left(47^{ 5 }\right) \\ r_{ 5 } &= 16243678 a + 29109024 +O\left(47^{ 5 }\right) \\ r_{ 6 } &= -36573485 +O\left(47^{ 5 }\right) \\ r_{ 7 } &= -16243678 a - 13821465 +O\left(47^{ 5 }\right) \\ r_{ 8 } &= 104151771 +O\left(47^{ 5 }\right) \\ \end{aligned}

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(1,5)(2,4)(3,7)(6,8)$ $(1,8)(3,7)(5,6)$ $(1,6,5,8)(2,7,4,3)$ $(1,6,3)(5,8,7)$ $(1,2,5,4)(3,8,7,6)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,5)(2,4)(3,7)(6,8)$ $-2$ $12$ $2$ $(1,8)(3,7)(5,6)$ $0$ $8$ $3$ $(1,8,4)(2,5,6)$ $-1$ $6$ $4$ $(1,2,5,4)(3,8,7,6)$ $0$ $8$ $6$ $(1,2,8,5,4,6)(3,7)$ $1$ $6$ $8$ $(1,2,6,7,5,4,8,3)$ $-\zeta_{8}^{3} - \zeta_{8}$ $6$ $8$ $(1,4,6,3,5,2,8,7)$ $\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.