# Properties

 Label 4.4549.5t5.1c1 Dimension 4 Group $S_5$ Conductor $4549$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $4549$ Artin number field: Splitting field of $f= x^{5} - x^{4} + 2 x^{3} - 2 x^{2} - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even Determinant: 1.4549.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $14 + 38\cdot 73 + 40\cdot 73^{2} + 39\cdot 73^{3} + 34\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 2 }$ $=$ $29 + 35\cdot 73 + 34\cdot 73^{2} + 41\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 3 }$ $=$ $43 + 44\cdot 73 + 5\cdot 73^{2} + 21\cdot 73^{3} + 66\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 4 }$ $=$ $63 + 64\cdot 73 + 25\cdot 73^{2} + 63\cdot 73^{3} + 46\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 5 }$ $=$ $71 + 35\cdot 73 + 39\cdot 73^{2} + 21\cdot 73^{3} + 30\cdot 73^{4} +O\left(73^{ 5 }\right)$

### 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.