# Properties

 Label 3.52441.6t8.c.a Dimension $3$ Group $S_4$ Conductor $52441$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $$52441$$$$\medspace = 229^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.229.1 Galois orbit size: $1$ Smallest permutation container: $S_4$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.0.229.1

## Defining polynomial

 $f(x)$ $=$ $x^{4} - x + 1$.

The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $135 + 139\cdot 193 + 151\cdot 193^{2} + 188\cdot 193^{3} + 63\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 2 }$ $=$ $145 + 66\cdot 193 + 184\cdot 193^{2} + 103\cdot 193^{3} + 30\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 3 }$ $=$ $148 + 174\cdot 193 + 160\cdot 193^{2} + 29\cdot 193^{3} + 160\cdot 193^{4} +O\left(193^{ 5 }\right)$ $r_{ 4 }$ $=$ $151 + 4\cdot 193 + 82\cdot 193^{2} + 63\cdot 193^{3} + 131\cdot 193^{4} +O\left(193^{ 5 }\right)$

## Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,2,3,4)$ $(1,2)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,4)$ $-1$ $6$ $2$ $(1,2)$ $-1$ $8$ $3$ $(1,2,3)$ $0$ $6$ $4$ $(1,2,3,4)$ $1$

The blue line marks the conjugacy class containing complex conjugation.