# Properties

 Label 4.2617.5t5.b Dimension 4 Group $S_5$ Conductor $2617$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $2617$ 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: Even Projective image: $S_5$ Projective field: Galois closure of 5.1.2617.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{2} + 24 x + 2$
Roots:
 $r_{ 1 }$ $=$ $11 a + 13 + \left(16 a + 17\right)\cdot 29 + \left(12 a + 18\right)\cdot 29^{2} + \left(21 a + 28\right)\cdot 29^{3} + \left(4 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 2 }$ $=$ $18 a + 10 + \left(12 a + 1\right)\cdot 29 + \left(16 a + 7\right)\cdot 29^{2} + \left(7 a + 7\right)\cdot 29^{3} + \left(24 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 3 }$ $=$ $20 a + 20 + \left(10 a + 21\right)\cdot 29 + \left(27 a + 26\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(14 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 4 }$ $=$ $9 a + 4 + \left(18 a + 26\right)\cdot 29 + \left(a + 7\right)\cdot 29^{2} + \left(18 a + 9\right)\cdot 29^{3} + \left(14 a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 + 20\cdot 29 + 26\cdot 29^{2} + 29^{3} + 16\cdot 29^{4} +O\left(29^{ 5 }\right)$

### 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.