# Properties

 Label 4.53_61.5t5.1 Dimension 4 Group $S_5$ Conductor $53 \cdot 61$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $S_5$ Conductor: $3233= 53 \cdot 61$ Artin number field: Splitting field of $f= x^{5} - x^{2} - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_5$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 383 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $88 + 282\cdot 383 + 41\cdot 383^{2} + 170\cdot 383^{3} + 220\cdot 383^{4} +O\left(383^{ 5 }\right)$ $r_{ 2 }$ $=$ $89 + 155\cdot 383 + 93\cdot 383^{2} + 241\cdot 383^{3} + 206\cdot 383^{4} +O\left(383^{ 5 }\right)$ $r_{ 3 }$ $=$ $120 + 178\cdot 383 + 256\cdot 383^{2} + 144\cdot 383^{3} + 377\cdot 383^{4} +O\left(383^{ 5 }\right)$ $r_{ 4 }$ $=$ $127 + 109\cdot 383 + 48\cdot 383^{2} + 292\cdot 383^{3} + 181\cdot 383^{4} +O\left(383^{ 5 }\right)$ $r_{ 5 }$ $=$ $342 + 40\cdot 383 + 326\cdot 383^{2} + 300\cdot 383^{3} + 162\cdot 383^{4} +O\left(383^{ 5 }\right)$

### 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.