# Properties

 Label 2.72075.24t22.a.a Dimension $2$ Group $\textrm{GL(2,3)}$ Conductor $72075$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $$72075$$$$\medspace = 3 \cdot 5^{2} \cdot 31^{2}$$ Artin stem field: Galois closure of 8.2.15584416875.1 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 4.2.72075.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 4x^{7} + 7x^{6} - 7x^{5} - 11x^{4} + 29x^{3} - 40x^{2} + 25x - 25$$ x^8 - 4*x^7 + 7*x^6 - 7*x^5 - 11*x^4 + 29*x^3 - 40*x^2 + 25*x - 25 .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

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

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.