# Properties

 Label 5.6417228161287.6t14.a Dimension 5 Group $S_5$ Conductor $18583^{3}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $6417228161287= 18583^{3}$ Artin number field: Splitting field of $f= x^{5} - x^{4} - x^{3} - 2 x^{2} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PGL(2,5)$ Parity: Odd Projective image: $S_5$ Projective field: Galois closure of 5.3.18583.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $x^{2} + 6 x + 3$
Roots:
 $r_{ 1 }$ $=$ $2 a + 6 + \left(2 a + 1\right)\cdot 7 + \left(2 a + 6\right)\cdot 7^{2} + 4 a\cdot 7^{3} + 5\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 2 }$ $=$ $2 a + 4 + 4 a\cdot 7 + \left(4 a + 3\right)\cdot 7^{2} + \left(3 a + 1\right)\cdot 7^{3} + \left(4 a + 1\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 3 }$ $=$ $5 a + 1 + \left(4 a + 2\right)\cdot 7 + \left(4 a + 6\right)\cdot 7^{2} + \left(2 a + 2\right)\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 4 }$ $=$ $5 + 6\cdot 7 + 7^{2} + 7^{3} + 4\cdot 7^{4} +O\left(7^{ 5 }\right)$ $r_{ 5 }$ $=$ $5 a + 6 + \left(2 a + 2\right)\cdot 7 + \left(2 a + 3\right)\cdot 7^{2} + 3 a\cdot 7^{3} + \left(2 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character values $c1$ $1$ $1$ $()$ $5$ $10$ $2$ $(1,2)$ $-1$ $15$ $2$ $(1,2)(3,4)$ $1$ $20$ $3$ $(1,2,3)$ $-1$ $30$ $4$ $(1,2,3,4)$ $1$ $24$ $5$ $(1,2,3,4,5)$ $0$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.