# Properties

 Label 5.3196661779891.6t16.a.a Dimension 5 Group $S_6$ Conductor $14731^{3}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $5$ Group: $S_6$ Conductor: $3196661779891= 14731^{3}$ Artin number field: Splitting field of 6.0.14731.1 defined by $f= x^{6} - x^{5} + x^{3} - x^{2} + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_6$ Parity: Odd Determinant: 1.14731.2t1.a.a Projective image: $S_6$ Projective field: Galois closure of 6.0.14731.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots:
 $r_{ 1 }$ $=$ $21 + 42\cdot 47 + 46\cdot 47^{2} + 10\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 + 18\cdot 47 + 28\cdot 47^{2} + 9\cdot 47^{3} + 6\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $15 a + 10 + \left(29 a + 17\right)\cdot 47 + \left(46 a + 29\right)\cdot 47^{2} + \left(12 a + 40\right)\cdot 47^{3} + \left(38 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $42 a + 11 + \left(7 a + 14\right)\cdot 47 + \left(8 a + 14\right)\cdot 47^{2} + \left(45 a + 12\right)\cdot 47^{3} + \left(26 a + 30\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 5 }$ $=$ $32 a + 40 + \left(17 a + 13\right)\cdot 47 + 46\cdot 47^{2} + \left(34 a + 19\right)\cdot 47^{3} + \left(8 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 6 }$ $=$ $5 a + 1 + \left(39 a + 35\right)\cdot 47 + \left(38 a + 22\right)\cdot 47^{2} + a\cdot 47^{3} + \left(20 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

### 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 value $1$ $1$ $()$ $5$ $15$ $2$ $(1,2)(3,4)(5,6)$ $3$ $15$ $2$ $(1,2)$ $-1$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $2$ $40$ $3$ $(1,2,3)$ $-1$ $90$ $4$ $(1,2,3,4)(5,6)$ $-1$ $90$ $4$ $(1,2,3,4)$ $1$ $144$ $5$ $(1,2,3,4,5)$ $0$ $120$ $6$ $(1,2,3,4,5,6)$ $0$ $120$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.