# Properties

 Label 2.3204.24t22.d.b Dimension $2$ Group $\textrm{GL(2,3)}$ Conductor $3204$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $$3204$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 89$$ Artin stem field: Galois closure of 8.2.24668275248.4 Galois orbit size: $2$ Smallest permutation container: 24T22 Parity: odd Determinant: 1.267.2t1.a.a Projective image: $S_4$ Projective stem field: Galois closure of 4.2.9612.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{7} + 10x^{6} - 7x^{5} + 16x^{4} - 13x^{3} - 23x^{2} + 74x + 16$$ x^8 - x^7 + 10*x^6 - 7*x^5 + 16*x^4 - 13*x^3 - 23*x^2 + 74*x + 16 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$10 a + 3 + \left(2 a + 8\right)\cdot 11 + 7 a\cdot 11^{2} + \left(8 a + 7\right)\cdot 11^{3} + \left(7 a + 2\right)\cdot 11^{4} + \left(2 a + 9\right)\cdot 11^{5} + \left(3 a + 6\right)\cdot 11^{6} + \left(a + 6\right)\cdot 11^{7} + \left(3 a + 6\right)\cdot 11^{8} +O(11^{9})$$ 10*a + 3 + (2*a + 8)*11 + 7*a*11^2 + (8*a + 7)*11^3 + (7*a + 2)*11^4 + (2*a + 9)*11^5 + (3*a + 6)*11^6 + (a + 6)*11^7 + (3*a + 6)*11^8+O(11^9) $r_{ 2 }$ $=$ $$8 a + 7 + \left(6 a + 1\right)\cdot 11 + \left(7 a + 2\right)\cdot 11^{2} + \left(6 a + 3\right)\cdot 11^{3} + \left(8 a + 8\right)\cdot 11^{4} + \left(9 a + 10\right)\cdot 11^{5} + \left(7 a + 9\right)\cdot 11^{6} + \left(9 a + 6\right)\cdot 11^{7} + 2 a\cdot 11^{8} +O(11^{9})$$ 8*a + 7 + (6*a + 1)*11 + (7*a + 2)*11^2 + (6*a + 3)*11^3 + (8*a + 8)*11^4 + (9*a + 10)*11^5 + (7*a + 9)*11^6 + (9*a + 6)*11^7 + 2*a*11^8+O(11^9) $r_{ 3 }$ $=$ $$4 + 2\cdot 11^{2} + 4\cdot 11^{3} + 7\cdot 11^{4} + 2\cdot 11^{5} + 9\cdot 11^{6} + 7\cdot 11^{7} +O(11^{9})$$ 4 + 2*11^2 + 4*11^3 + 7*11^4 + 2*11^5 + 9*11^6 + 7*11^7+O(11^9) $r_{ 4 }$ $=$ $$5 a + 9 + \left(2 a + 1\right)\cdot 11 + \left(2 a + 9\right)\cdot 11^{2} + \left(4 a + 6\right)\cdot 11^{3} + \left(2 a + 7\right)\cdot 11^{4} + 10\cdot 11^{5} + \left(6 a + 3\right)\cdot 11^{6} + \left(10 a + 3\right)\cdot 11^{7} + 11^{8} +O(11^{9})$$ 5*a + 9 + (2*a + 1)*11 + (2*a + 9)*11^2 + (4*a + 6)*11^3 + (2*a + 7)*11^4 + 10*11^5 + (6*a + 3)*11^6 + (10*a + 3)*11^7 + 11^8+O(11^9) $r_{ 5 }$ $=$ $$3 a + 6 + \left(4 a + 9\right)\cdot 11 + \left(3 a + 3\right)\cdot 11^{2} + 4 a\cdot 11^{3} + \left(2 a + 3\right)\cdot 11^{4} + \left(a + 8\right)\cdot 11^{5} + \left(3 a + 9\right)\cdot 11^{6} + \left(a + 4\right)\cdot 11^{7} + \left(8 a + 2\right)\cdot 11^{8} +O(11^{9})$$ 3*a + 6 + (4*a + 9)*11 + (3*a + 3)*11^2 + 4*a*11^3 + (2*a + 3)*11^4 + (a + 8)*11^5 + (3*a + 9)*11^6 + (a + 4)*11^7 + (8*a + 2)*11^8+O(11^9) $r_{ 6 }$ $=$ $$a + 10 + \left(8 a + 9\right)\cdot 11 + \left(3 a + 4\right)\cdot 11^{2} + \left(2 a + 1\right)\cdot 11^{3} + \left(3 a + 3\right)\cdot 11^{4} + \left(8 a + 1\right)\cdot 11^{5} + \left(7 a + 6\right)\cdot 11^{6} + \left(9 a + 8\right)\cdot 11^{7} + \left(7 a + 6\right)\cdot 11^{8} +O(11^{9})$$ a + 10 + (8*a + 9)*11 + (3*a + 4)*11^2 + (2*a + 1)*11^3 + (3*a + 3)*11^4 + (8*a + 1)*11^5 + (7*a + 6)*11^6 + (9*a + 8)*11^7 + (7*a + 6)*11^8+O(11^9) $r_{ 7 }$ $=$ $$10 + 5\cdot 11 + 5\cdot 11^{2} + 10\cdot 11^{3} + 9\cdot 11^{4} + 2\cdot 11^{5} + 3\cdot 11^{6} + 10\cdot 11^{7} + 8\cdot 11^{8} +O(11^{9})$$ 10 + 5*11 + 5*11^2 + 10*11^3 + 9*11^4 + 2*11^5 + 3*11^6 + 10*11^7 + 8*11^8+O(11^9) $r_{ 8 }$ $=$ $$6 a + 7 + \left(8 a + 6\right)\cdot 11 + \left(8 a + 4\right)\cdot 11^{2} + \left(6 a + 10\right)\cdot 11^{3} + \left(8 a + 1\right)\cdot 11^{4} + \left(10 a + 9\right)\cdot 11^{5} + \left(4 a + 5\right)\cdot 11^{6} + 6\cdot 11^{7} + \left(10 a + 5\right)\cdot 11^{8} +O(11^{9})$$ 6*a + 7 + (8*a + 6)*11 + (8*a + 4)*11^2 + (6*a + 10)*11^3 + (8*a + 1)*11^4 + (10*a + 9)*11^5 + (4*a + 5)*11^6 + 6*11^7 + (10*a + 5)*11^8+O(11^9)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.