# Properties

 Label 5.67_743.6t16.1 Dimension 5 Group $S_6$ Conductor $67 \cdot 743$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $49781= 67 \cdot 743$ Artin number field: Splitting field of $f= x^{6} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_6$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{2} + 38 x + 6$
Roots:
 $r_{ 1 }$ $=$ $17 a + 25 + \left(27 a + 2\right)\cdot 41 + \left(a + 36\right)\cdot 41^{2} + \left(15 a + 32\right)\cdot 41^{3} + \left(5 a + 32\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $11 a + 36 + \left(22 a + 26\right)\cdot 41 + \left(34 a + 1\right)\cdot 41^{2} + \left(39 a + 6\right)\cdot 41^{3} + \left(30 a + 5\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $24 a + 35 + \left(13 a + 26\right)\cdot 41 + \left(39 a + 13\right)\cdot 41^{2} + \left(25 a + 35\right)\cdot 41^{3} + \left(35 a + 33\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $30 a + 28 + 18 a\cdot 41 + \left(6 a + 1\right)\cdot 41^{2} + \left(a + 9\right)\cdot 41^{3} + \left(10 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 5 }$ $=$ $24 a + 25 + \left(24 a + 28\right)\cdot 41 + \left(38 a + 30\right)\cdot 41^{2} + \left(19 a + 29\right)\cdot 41^{3} + \left(34 a + 36\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 6 }$ $=$ $17 a + 15 + \left(16 a + 37\right)\cdot 41 + \left(2 a + 39\right)\cdot 41^{2} + \left(21 a + 9\right)\cdot 41^{3} + \left(6 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $5$ $15$ $2$ $(1,2)(3,4)(5,6)$ $-1$ $15$ $2$ $(1,2)$ $3$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $-1$ $40$ $3$ $(1,2,3)$ $2$ $90$ $4$ $(1,2,3,4)(5,6)$ $-1$ $90$ $4$ $(1,2,3,4)$ $1$ $144$ $5$ $(1,2,3,4,5)$ $0$ $120$ $6$ $(1,2,3,4,5,6)$ $-1$ $120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.