# Properties

 Label 2.1444.24t22.a Dimension $2$ Group $\textrm{GL(2,3)}$ Conductor $1444$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $$1444$$$$\medspace = 2^{2} \cdot 19^{2}$$ Artin number field: Galois closure of 8.2.14301947824.1 Galois orbit size: $2$ Smallest permutation container: 24T22 Parity: odd Projective image: $S_4$ Projective field: 4.2.27436.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12 x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$2 + 7\cdot 13 + 11\cdot 13^{2} + 12\cdot 13^{3} + 6\cdot 13^{4} + 6\cdot 13^{5} + 7\cdot 13^{6} + 12\cdot 13^{7} + 7\cdot 13^{8} + 7\cdot 13^{9} +O(13^{10})$$ $r_{ 2 }$ $=$ $$3 a + 7 + \left(4 a + 4\right)\cdot 13 + \left(2 a + 12\right)\cdot 13^{2} + \left(3 a + 1\right)\cdot 13^{3} + \left(6 a + 2\right)\cdot 13^{4} + \left(11 a + 6\right)\cdot 13^{5} + \left(12 a + 7\right)\cdot 13^{6} + \left(12 a + 1\right)\cdot 13^{7} + 5\cdot 13^{8} + \left(12 a + 12\right)\cdot 13^{9} +O(13^{10})$$ $r_{ 3 }$ $=$ $$10 a + 7 + \left(8 a + 8\right)\cdot 13 + 10 a\cdot 13^{2} + \left(9 a + 11\right)\cdot 13^{3} + \left(6 a + 10\right)\cdot 13^{4} + \left(a + 6\right)\cdot 13^{5} + 5\cdot 13^{6} + 11\cdot 13^{7} + \left(12 a + 7\right)\cdot 13^{8} +O(13^{10})$$ $r_{ 4 }$ $=$ $$7 a + 10 + \left(11 a + 10\right)\cdot 13 + \left(a + 4\right)\cdot 13^{2} + \left(7 a + 10\right)\cdot 13^{3} + \left(3 a + 1\right)\cdot 13^{4} + \left(a + 1\right)\cdot 13^{5} + \left(3 a + 12\right)\cdot 13^{6} + \left(7 a + 10\right)\cdot 13^{7} + \left(8 a + 5\right)\cdot 13^{8} + \left(8 a + 6\right)\cdot 13^{9} +O(13^{10})$$ $r_{ 5 }$ $=$ $$10 a + 10 + \left(8 a + 5\right)\cdot 13 + \left(10 a + 10\right)\cdot 13^{2} + \left(9 a + 2\right)\cdot 13^{3} + \left(6 a + 5\right)\cdot 13^{4} + \left(a + 11\right)\cdot 13^{5} + 8\cdot 13^{6} + 13^{7} + \left(12 a + 6\right)\cdot 13^{8} + 10\cdot 13^{9} +O(13^{10})$$ $r_{ 6 }$ $=$ $$12 + 5\cdot 13 + 13^{2} + 6\cdot 13^{4} + 6\cdot 13^{5} + 5\cdot 13^{6} + 5\cdot 13^{8} + 5\cdot 13^{9} +O(13^{10})$$ $r_{ 7 }$ $=$ $$3 a + 4 + \left(4 a + 7\right)\cdot 13 + \left(2 a + 2\right)\cdot 13^{2} + \left(3 a + 10\right)\cdot 13^{3} + \left(6 a + 7\right)\cdot 13^{4} + \left(11 a + 1\right)\cdot 13^{5} + \left(12 a + 4\right)\cdot 13^{6} + \left(12 a + 11\right)\cdot 13^{7} + 6\cdot 13^{8} + \left(12 a + 2\right)\cdot 13^{9} +O(13^{10})$$ $r_{ 8 }$ $=$ $$6 a + 4 + \left(a + 2\right)\cdot 13 + \left(11 a + 8\right)\cdot 13^{2} + \left(5 a + 2\right)\cdot 13^{3} + \left(9 a + 11\right)\cdot 13^{4} + \left(11 a + 11\right)\cdot 13^{5} + 9 a\cdot 13^{6} + \left(5 a + 2\right)\cdot 13^{7} + \left(4 a + 7\right)\cdot 13^{8} + \left(4 a + 6\right)\cdot 13^{9} +O(13^{10})$$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,6)(2,3)(4,8)(5,7)$ $-2$ $-2$ $12$ $2$ $(1,6)(2,4)(3,8)$ $0$ $0$ $8$ $3$ $(1,3,7)(2,5,6)$ $-1$ $-1$ $6$ $4$ $(1,4,6,8)(2,5,3,7)$ $0$ $0$ $8$ $6$ $(1,5,3,6,7,2)(4,8)$ $1$ $1$ $6$ $8$ $(1,8,7,2,6,4,5,3)$ $-\zeta_{8}^{3} - \zeta_{8}$ $\zeta_{8}^{3} + \zeta_{8}$ $6$ $8$ $(1,4,7,3,6,8,5,2)$ $\zeta_{8}^{3} + \zeta_{8}$ $-\zeta_{8}^{3} - \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.