# Properties

 Label 2.37_59.24t22.1 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $37 \cdot 59$ Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $2183= 37 \cdot 59$ Artin number field: Splitting field of $f= x^{8} - x^{7} - 8 x^{6} + 15 x^{5} + 20 x^{4} - 46 x^{3} - 17 x^{2} + 74 x - 37$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $x^{2} + 78 x + 3$
Roots:
 $r_{ 1 }$ $=$ $6 + 33\cdot 79 + 23\cdot 79^{2} + 61\cdot 79^{3} + 9\cdot 79^{4} + 27\cdot 79^{5} +O\left(79^{ 6 }\right)$ $r_{ 2 }$ $=$ $6 a + 77 + \left(36 a + 1\right)\cdot 79 + \left(23 a + 17\right)\cdot 79^{2} + \left(64 a + 39\right)\cdot 79^{3} + \left(75 a + 75\right)\cdot 79^{4} + \left(25 a + 56\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ $r_{ 3 }$ $=$ $35 a + 17 + \left(18 a + 17\right)\cdot 79 + \left(6 a + 41\right)\cdot 79^{2} + \left(17 a + 51\right)\cdot 79^{3} + \left(42 a + 74\right)\cdot 79^{4} + \left(42 a + 59\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ $r_{ 4 }$ $=$ $51 a + 27 + \left(68 a + 32\right)\cdot 79 + \left(33 a + 59\right)\cdot 79^{2} + \left(54 a + 64\right)\cdot 79^{3} + \left(66 a + 51\right)\cdot 79^{4} + \left(75 a + 59\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ $r_{ 5 }$ $=$ $73 a + 4 + \left(42 a + 32\right)\cdot 79 + \left(55 a + 4\right)\cdot 79^{2} + \left(14 a + 1\right)\cdot 79^{3} + \left(3 a + 8\right)\cdot 79^{4} + \left(53 a + 7\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ $r_{ 6 }$ $=$ $44 a + 52 + 60 a\cdot 79 + \left(72 a + 29\right)\cdot 79^{2} + \left(61 a + 62\right)\cdot 79^{3} + \left(36 a + 20\right)\cdot 79^{4} + \left(36 a + 60\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ $r_{ 7 }$ $=$ $56 + 69\cdot 79 + 37\cdot 79^{2} + 29\cdot 79^{3} + 11\cdot 79^{4} + 55\cdot 79^{5} +O\left(79^{ 6 }\right)$ $r_{ 8 }$ $=$ $28 a + 78 + \left(10 a + 49\right)\cdot 79 + \left(45 a + 24\right)\cdot 79^{2} + \left(24 a + 6\right)\cdot 79^{3} + \left(12 a + 64\right)\cdot 79^{4} + \left(3 a + 68\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$

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

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

### Character values on conjugacy classes

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