# Properties

 Label 5.33792250337.6t14.a.a Dimension $5$ Group $S_5$ Conductor $33792250337$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $5$ Group: $S_5$ Conductor: $$33792250337$$$$\medspace = 53^{3} \cdot 61^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 5.1.3233.1 Galois orbit size: $1$ Smallest permutation container: $\PGL(2,5)$ Parity: even Determinant: 1.3233.2t1.a.a Projective image: $S_5$ Projective field: Galois closure of 5.1.3233.1

## Defining polynomial

 $f(x)$ $=$ $x^{5} - x^{2} - 1$.

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 value $1$ $1$ $()$ $5$ $10$ $2$ $(1,2)$ $-1$ $15$ $2$ $(1,2)(3,4)$ $1$ $20$ $3$ $(1,2,3)$ $-1$ $30$ $4$ $(1,2,3,4)$ $1$ $24$ $5$ $(1,2,3,4,5)$ $0$ $20$ $6$ $(1,2,3)(4,5)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.