# Properties

 Label 4.2e3_887.5t5.1c1 Dimension 4 Group $S_5$ Conductor $2^{3} \cdot 887$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $7096= 2^{3} \cdot 887$ Artin number field: Splitting field of $f= x^{5} - x^{2} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Determinant: 1.2e3_887.2t1.1c1

## 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:
 $r_{ 1 }$ $=$ $16 a + 2 + \left(5 a + 8\right)\cdot 19 + \left(8 a + 8\right)\cdot 19^{2} + \left(18 a + 1\right)\cdot 19^{3} + \left(10 a + 12\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 a + 18 + \left(13 a + 16\right)\cdot 19 + \left(10 a + 10\right)\cdot 19^{2} + 11\cdot 19^{3} + \left(8 a + 4\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $14 + 13\cdot 19 + 17\cdot 19^{2} + 14\cdot 19^{3} + 15\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $5 a + 9 + \left(a + 1\right)\cdot 19 + \left(12 a + 14\right)\cdot 19^{2} + \left(16 a + 2\right)\cdot 19^{3} + \left(a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $14 a + 14 + \left(17 a + 16\right)\cdot 19 + \left(6 a + 5\right)\cdot 19^{2} + \left(2 a + 7\right)\cdot 19^{3} + \left(17 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

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