# Properties

 Label 4.3089.5t5.1c1 Dimension 4 Group $S_5$ Conductor $3089$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $3089$ Artin number field: Splitting field of $f= x^{5} - x^{3} + 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Determinant: 1.3089.2t1.1c1

## Galois action

### Roots of defining polynomial

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

### 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.