# Properties

 Label 5.73_709.6t16.1 Dimension 5 Group $S_6$ Conductor $73 \cdot 709$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $51757= 73 \cdot 709$ Artin number field: Splitting field of $f= x^{6} - 2 x^{3} - 2 x^{2} + x + 1$ 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 $\Q_{ 113 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $x^{2} + 101 x + 3$
Roots:
 $r_{ 1 }$ $=$ $23 a + 80 + \left(43 a + 75\right)\cdot 113 + \left(104 a + 69\right)\cdot 113^{2} + \left(a + 98\right)\cdot 113^{3} + \left(29 a + 104\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ $r_{ 2 }$ $=$ $74 a + 93 + \left(81 a + 50\right)\cdot 113 + \left(61 a + 28\right)\cdot 113^{2} + \left(101 a + 41\right)\cdot 113^{3} + \left(7 a + 22\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ $r_{ 3 }$ $=$ $13 a + 71 + \left(5 a + 52\right)\cdot 113 + \left(54 a + 58\right)\cdot 113^{2} + \left(96 a + 13\right)\cdot 113^{3} + \left(96 a + 74\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ $r_{ 4 }$ $=$ $39 a + 77 + \left(31 a + 52\right)\cdot 113 + \left(51 a + 9\right)\cdot 113^{2} + \left(11 a + 68\right)\cdot 113^{3} + \left(105 a + 15\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ $r_{ 5 }$ $=$ $90 a + 17 + \left(69 a + 6\right)\cdot 113 + \left(8 a + 36\right)\cdot 113^{2} + \left(111 a + 17\right)\cdot 113^{3} + \left(83 a + 112\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$ $r_{ 6 }$ $=$ $100 a + 1 + \left(107 a + 101\right)\cdot 113 + \left(58 a + 23\right)\cdot 113^{2} + \left(16 a + 100\right)\cdot 113^{3} + \left(16 a + 9\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$

### 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.