# Properties

 Label 2.37_59.24t22.3c2 Dimension 2 Group $\textrm{GL(2,3)}$ Conductor $37 \cdot 59$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $2$ Group: $\textrm{GL(2,3)}$ Conductor: $2183= 37 \cdot 59$ Artin number field: Splitting field of $f= x^{8} - x^{7} - 5 x^{6} - x^{5} + 12 x^{4} + 18 x^{3} - 15 x^{2} - 27 x + 4$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: 24T22 Parity: Odd Determinant: 1.37_59.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $x^{2} + 78 x + 3$
Roots:
 $r_{ 1 }$ $=$ $36 + 57\cdot 79 + 33\cdot 79^{2} + 40\cdot 79^{3} + 42\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 a + 51 + \left(a + 2\right)\cdot 79 + \left(7 a + 39\right)\cdot 79^{2} + \left(a + 38\right)\cdot 79^{3} + \left(12 a + 52\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 3 }$ $=$ $26 + 45\cdot 79 + 27\cdot 79^{2} + 50\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 4 }$ $=$ $66 a + 35 + \left(77 a + 27\right)\cdot 79 + \left(34 a + 4\right)\cdot 79^{2} + \left(47 a + 50\right)\cdot 79^{3} + \left(76 a + 13\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 5 }$ $=$ $54 a + 59 + \left(55 a + 70\right)\cdot 79 + \left(73 a + 53\right)\cdot 79^{2} + \left(33 a + 40\right)\cdot 79^{3} + \left(41 a + 57\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 6 }$ $=$ $76 a + 54 + 77 a\cdot 79 + \left(71 a + 45\right)\cdot 79^{2} + \left(77 a + 32\right)\cdot 79^{3} + \left(66 a + 63\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 7 }$ $=$ $13 a + 22 + \left(a + 39\right)\cdot 79 + \left(44 a + 40\right)\cdot 79^{2} + \left(31 a + 62\right)\cdot 79^{3} + \left(2 a + 42\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 8 }$ $=$ $25 a + 34 + \left(23 a + 72\right)\cdot 79 + \left(5 a + 71\right)\cdot 79^{2} + 45 a\cdot 79^{3} + \left(37 a + 65\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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