# Properties

 Label 2.3267.24t22.b.b Dimension $2$ Group $\textrm{GL(2,3)}$ Conductor $3267$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $$3267$$$$\medspace = 3^{3} \cdot 11^{2}$$ Artin stem field: Galois closure of 8.2.32019867.2 Galois orbit size: $2$ Smallest permutation container: 24T22 Parity: odd Determinant: 1.3.2t1.a.a Projective image: $S_4$ Projective stem field: Galois closure of 4.2.3267.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{7} + x^{6} - 2x^{5} - x^{4} - 2x^{3} + x^{2} - x + 1$$ x^8 - x^7 + x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 - x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$10 a + 27 + \left(19 a + 14\right)\cdot 29 + \left(23 a + 15\right)\cdot 29^{2} + \left(16 a + 12\right)\cdot 29^{3} + \left(8 a + 14\right)\cdot 29^{4} + \left(23 a + 8\right)\cdot 29^{5} +O(29^{6})$$ 10*a + 27 + (19*a + 14)*29 + (23*a + 15)*29^2 + (16*a + 12)*29^3 + (8*a + 14)*29^4 + (23*a + 8)*29^5+O(29^6) $r_{ 2 }$ $=$ $$4 a + 5 + \left(13 a + 14\right)\cdot 29 + \left(26 a + 25\right)\cdot 29^{2} + \left(4 a + 11\right)\cdot 29^{3} + \left(6 a + 14\right)\cdot 29^{4} + \left(7 a + 28\right)\cdot 29^{5} +O(29^{6})$$ 4*a + 5 + (13*a + 14)*29 + (26*a + 25)*29^2 + (4*a + 11)*29^3 + (6*a + 14)*29^4 + (7*a + 28)*29^5+O(29^6) $r_{ 3 }$ $=$ $$24 a + 17 + \left(5 a + 12\right)\cdot 29 + \left(23 a + 14\right)\cdot 29^{2} + 17\cdot 29^{3} + \left(7 a + 27\right)\cdot 29^{4} + \left(5 a + 27\right)\cdot 29^{5} +O(29^{6})$$ 24*a + 17 + (5*a + 12)*29 + (23*a + 14)*29^2 + 17*29^3 + (7*a + 27)*29^4 + (5*a + 27)*29^5+O(29^6) $r_{ 4 }$ $=$ $$7 + 2\cdot 29 + 25\cdot 29^{2} + 5\cdot 29^{3} + 4\cdot 29^{4} + 17\cdot 29^{5} +O(29^{6})$$ 7 + 2*29 + 25*29^2 + 5*29^3 + 4*29^4 + 17*29^5+O(29^6) $r_{ 5 }$ $=$ $$25 + 21\cdot 29 + 28\cdot 29^{2} + 15\cdot 29^{3} + 28\cdot 29^{4} + 14\cdot 29^{5} +O(29^{6})$$ 25 + 21*29 + 28*29^2 + 15*29^3 + 28*29^4 + 14*29^5+O(29^6) $r_{ 6 }$ $=$ $$5 a + 21 + \left(23 a + 17\right)\cdot 29 + \left(5 a + 8\right)\cdot 29^{2} + \left(28 a + 27\right)\cdot 29^{3} + \left(21 a + 3\right)\cdot 29^{4} + \left(23 a + 18\right)\cdot 29^{5} +O(29^{6})$$ 5*a + 21 + (23*a + 17)*29 + (5*a + 8)*29^2 + (28*a + 27)*29^3 + (21*a + 3)*29^4 + (23*a + 18)*29^5+O(29^6) $r_{ 7 }$ $=$ $$19 a + 19 + \left(9 a + 14\right)\cdot 29 + \left(5 a + 27\right)\cdot 29^{2} + \left(12 a + 14\right)\cdot 29^{3} + \left(20 a + 11\right)\cdot 29^{4} + 5 a\cdot 29^{5} +O(29^{6})$$ 19*a + 19 + (9*a + 14)*29 + (5*a + 27)*29^2 + (12*a + 14)*29^3 + (20*a + 11)*29^4 + 5*a*29^5+O(29^6) $r_{ 8 }$ $=$ $$25 a + 25 + \left(15 a + 17\right)\cdot 29 + \left(2 a + 28\right)\cdot 29^{2} + \left(24 a + 9\right)\cdot 29^{3} + \left(22 a + 11\right)\cdot 29^{4} + 21 a\cdot 29^{5} +O(29^{6})$$ 25*a + 25 + (15*a + 17)*29 + (2*a + 28)*29^2 + (24*a + 9)*29^3 + (22*a + 11)*29^4 + 21*a*29^5+O(29^6)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $12$ $2$ $(1,5)(2,7)(4,8)$ $0$ $8$ $3$ $(1,7,4)(2,5,8)$ $-1$ $6$ $4$ $(1,4,8,5)(2,6,7,3)$ $0$ $8$ $6$ $(1,2,4,8,7,5)(3,6)$ $1$ $6$ $8$ $(1,6,4,7,8,3,5,2)$ $\zeta_{8}^{3} + \zeta_{8}$ $6$ $8$ $(1,3,4,2,8,6,5,7)$ $-\zeta_{8}^{3} - \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.