# Properties

 Label 2.127449.24t22.a.a Dimension $2$ Group $\textrm{GL(2,3)}$ Conductor $127449$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $$127449$$$$\medspace = 3^{2} \cdot 7^{2} \cdot 17^{2}$$ Artin stem field: Galois closure of 8.2.438567685227.1 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 4.2.7803.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} + 3x^{6} - 9x^{5} + 15x^{4} - 66x^{3} + 147x^{2} - 165x - 3$$ x^8 + 3*x^6 - 9*x^5 + 15*x^4 - 66*x^3 + 147*x^2 - 165*x - 3 .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.

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

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.