# Properties

 Label 3.3_457.4t5.1c1 Dimension 3 Group $S_4$ Conductor $3 \cdot 457$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $1371= 3 \cdot 457$ Artin number field: Splitting field of $f= x^{4} + 2 x^{2} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Odd Determinant: 1.3_457.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $7 + 23\cdot 53 + 23\cdot 53^{2} + 25\cdot 53^{3} + 30\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 2 }$ $=$ $13 + 41\cdot 53 + 25\cdot 53^{2} + 39\cdot 53^{3} + 12\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 3 }$ $=$ $42 + 13\cdot 53^{2} + 42\cdot 53^{3} + 49\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 4 }$ $=$ $44 + 40\cdot 53 + 43\cdot 53^{2} + 51\cdot 53^{3} + 12\cdot 53^{4} +O\left(53^{ 5 }\right)$

### 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.