# Properties

 Label 4.4817.5t5.a Dimension 4 Group $S_5$ Conductor $4817$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $4817$ Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 2 x^{3} - x^{2} + 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Projective image: $S_5$ Projective field: Galois closure of 5.1.4817.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $16 + 12\cdot 19 + 6\cdot 19^{2} + 9\cdot 19^{3} + 7\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $13 + 9\cdot 19 + 16\cdot 19^{2} + 11\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $6 a + \left(2 a + 14\right)\cdot 19 + \left(6 a + 12\right)\cdot 19^{2} + \left(18 a + 8\right)\cdot 19^{3} + \left(14 a + 15\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $13 a + 6 + \left(16 a + 10\right)\cdot 19 + \left(12 a + 16\right)\cdot 19^{2} + 19^{3} + \left(4 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $5 + 10\cdot 19 + 4\cdot 19^{2} + 17\cdot 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right)$

### 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 $c1$ $1$ $1$ $()$ $4$ $10$ $2$ $(1,2)$ $2$ $15$ $2$ $(1,2)(3,4)$ $0$ $20$ $3$ $(1,2,3)$ $1$ $30$ $4$ $(1,2,3,4)$ $0$ $24$ $5$ $(1,2,3,4,5)$ $-1$ $20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.