# Properties

 Label 4.11e2_233e2.8t23.4c1 Dimension 4 Group $\textrm{GL(2,3)}$ Conductor $11^{2} \cdot 233^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $\textrm{GL(2,3)}$ Conductor: $6568969= 11^{2} \cdot 233^{2}$ Artin number field: Splitting field of $f= x^{8} - x^{7} - 3 x^{6} + x^{5} + 8 x^{4} + 2 x^{3} - 17 x^{2} - 49 x + 41$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\textrm{GL(2,3)}$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $x^{2} + 63 x + 2$
Roots:
 $r_{ 1 }$ $=$ $55 a + 64 + \left(30 a + 8\right)\cdot 67 + \left(4 a + 57\right)\cdot 67^{2} + \left(13 a + 12\right)\cdot 67^{3} + \left(11 a + 22\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 2 }$ $=$ $64 a + 54 + \left(51 a + 43\right)\cdot 67 + \left(11 a + 30\right)\cdot 67^{2} + \left(18 a + 53\right)\cdot 67^{3} + \left(38 a + 11\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 3 }$ $=$ $25 + 49\cdot 67 + 43\cdot 67^{2} + 22\cdot 67^{3} + 3\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 4 }$ $=$ $3 a + 42 + \left(15 a + 53\right)\cdot 67 + \left(55 a + 25\right)\cdot 67^{2} + \left(48 a + 47\right)\cdot 67^{3} + \left(28 a + 12\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 5 }$ $=$ $51 a + 25 + \left(58 a + 54\right)\cdot 67 + \left(63 a + 5\right)\cdot 67^{2} + \left(19 a + 42\right)\cdot 67^{3} + \left(28 a + 9\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 6 }$ $=$ $15 + 10\cdot 67 + 59\cdot 67^{2} + 37\cdot 67^{3} + 51\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 7 }$ $=$ $12 a + 16 + \left(36 a + 10\right)\cdot 67 + \left(62 a + 44\right)\cdot 67^{2} + \left(53 a + 60\right)\cdot 67^{3} + \left(55 a + 53\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 8 }$ $=$ $16 a + 28 + \left(8 a + 37\right)\cdot 67 + \left(3 a + 1\right)\cdot 67^{2} + \left(47 a + 58\right)\cdot 67^{3} + \left(38 a + 35\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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