# Properties

 Label 4.1609.5t5.1c1 Dimension 4 Group $S_5$ Conductor $1609$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $1609$ Artin number field: Splitting field of $f=x^{5} - x^{3} - x^{2} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: 5T5 Parity: Even Determinant: 1.1609.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$
Roots: \begin{aligned} r_{ 1 } &= 9220153 +O\left(31^{ 5 }\right) \\ r_{ 2 } &= -13897449 a + 537036 +O\left(31^{ 5 }\right) \\ r_{ 3 } &= 13897449 a + 2754943 +O\left(31^{ 5 }\right) \\ r_{ 4 } &= -6558550 a + 13161563 +O\left(31^{ 5 }\right) \\ r_{ 5 } &= 6558550 a + 2955456 +O\left(31^{ 5 }\right) \\ \end{aligned}

### 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.