# Properties

 Label 5.2e6_587.6t16.1 Dimension 5 Group $S_6$ Conductor $2^{6} \cdot 587$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $37568= 2^{6} \cdot 587$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} - x^{2} + 2 x - 1$ 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 $\Q_{ 167 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $x^{2} + 166 x + 5$
Roots:
 $r_{ 1 }$ $=$ $21 a + 3 + \left(150 a + 82\right)\cdot 167 + \left(89 a + 154\right)\cdot 167^{2} + \left(128 a + 73\right)\cdot 167^{3} + \left(142 a + 128\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 2 }$ $=$ $85 a + 8 + \left(67 a + 69\right)\cdot 167 + \left(33 a + 122\right)\cdot 167^{2} + \left(97 a + 61\right)\cdot 167^{3} + \left(128 a + 31\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 3 }$ $=$ $146 a + 24 + \left(16 a + 44\right)\cdot 167 + \left(77 a + 94\right)\cdot 167^{2} + \left(38 a + 112\right)\cdot 167^{3} + \left(24 a + 142\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 4 }$ $=$ $31 a + 5 + \left(60 a + 29\right)\cdot 167 + \left(104 a + 82\right)\cdot 167^{2} + \left(106 a + 62\right)\cdot 167^{3} + \left(58 a + 8\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 5 }$ $=$ $82 a + 93 + \left(99 a + 51\right)\cdot 167 + \left(133 a + 88\right)\cdot 167^{2} + \left(69 a + 125\right)\cdot 167^{3} + \left(38 a + 62\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 6 }$ $=$ $136 a + 36 + \left(106 a + 58\right)\cdot 167 + \left(62 a + 126\right)\cdot 167^{2} + \left(60 a + 64\right)\cdot 167^{3} + \left(108 a + 127\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$

### 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.