# Properties

 Label 4.13_347.5t5.1 Dimension 4 Group $S_5$ Conductor $13 \cdot 347$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $4511= 13 \cdot 347$ Artin number field: Splitting field of $f= x^{5} - x^{3} - 2 x^{2} + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Odd

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$
Roots:
 $r_{ 1 }$ $=$ $7 a + 16 + \left(5 a + 23\right)\cdot 37 + \left(20 a + 6\right)\cdot 37^{2} + \left(35 a + 36\right)\cdot 37^{3} + \left(22 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $21 + 6\cdot 37 + 6\cdot 37^{2} + 5\cdot 37^{3} + 27\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $18 a + 16 + \left(6 a + 36\right)\cdot 37 + \left(10 a + 27\right)\cdot 37^{2} + \left(8 a + 36\right)\cdot 37^{3} + \left(11 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $30 a + 7 + 31 a\cdot 37 + \left(16 a + 8\right)\cdot 37^{2} + \left(a + 10\right)\cdot 37^{3} + \left(14 a + 8\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 5 }$ $=$ $19 a + 14 + \left(30 a + 7\right)\cdot 37 + \left(26 a + 25\right)\cdot 37^{2} + \left(28 a + 22\right)\cdot 37^{3} + \left(25 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

### 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.