# Properties

 Label 3.316969.6t8.b.a Dimension $3$ Group $S_4$ Conductor $316969$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $$316969$$$$\medspace = 563^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 4.2.563.1 Galois orbit size: $1$ Smallest permutation container: $S_4$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_4$ Projective stem field: 4.2.563.1

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} + x^{2} - x - 1$$  .

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$80 + 67\cdot 137 + 75\cdot 137^{2} + 127\cdot 137^{3} + 122\cdot 137^{4} +O(137^{5})$$ $r_{ 2 }$ $=$ $$108 + 29\cdot 137 + 103\cdot 137^{2} + 73\cdot 137^{3} + 31\cdot 137^{4} +O(137^{5})$$ $r_{ 3 }$ $=$ $$111 + 67\cdot 137 + 30\cdot 137^{2} + 52\cdot 137^{3} + 26\cdot 137^{4} +O(137^{5})$$ $r_{ 4 }$ $=$ $$113 + 108\cdot 137 + 64\cdot 137^{2} + 20\cdot 137^{3} + 93\cdot 137^{4} +O(137^{5})$$

## 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.