# Properties

 Label 2.30276.24t22.a.a Dimension $2$ Group $\textrm{GL(2,3)}$ Conductor $30276$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $$30276$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 29^{2}$$ Artin stem field: Galois closure of 8.2.24749176752.1 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 4.2.90828.2

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 4x^{7} + 7x^{6} - 7x^{5} + 7x^{4} - 7x^{3} - 26x^{2} + 29x - 8$$ x^8 - 4*x^7 + 7*x^6 - 7*x^5 + 7*x^4 - 7*x^3 - 26*x^2 + 29*x - 8 .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21x + 5$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.