# Properties

 Label 4.7367.5t5.a Dimension 4 Group $S_5$ Conductor $53 \cdot 139$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $7367= 53 \cdot 139$ Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 3 x^{2} - 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Odd Projective image: $S_5$ Projective field: Galois closure of 5.3.7367.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 277 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $5 + 154\cdot 277 + 271\cdot 277^{2} + 170\cdot 277^{3} + 206\cdot 277^{4} +O\left(277^{ 5 }\right)$ $r_{ 2 }$ $=$ $145 + 54\cdot 277 + 47\cdot 277^{2} + 208\cdot 277^{3} + 90\cdot 277^{4} +O\left(277^{ 5 }\right)$ $r_{ 3 }$ $=$ $184 + 269\cdot 277 + 242\cdot 277^{2} + 209\cdot 277^{3} + 31\cdot 277^{4} +O\left(277^{ 5 }\right)$ $r_{ 4 }$ $=$ $228 + 192\cdot 277 + 114\cdot 277^{2} + 122\cdot 277^{3} + 272\cdot 277^{4} +O\left(277^{ 5 }\right)$ $r_{ 5 }$ $=$ $271 + 159\cdot 277 + 154\cdot 277^{2} + 119\cdot 277^{3} + 229\cdot 277^{4} +O\left(277^{ 5 }\right)$

### 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.