# Properties

 Label 3.2e4_53.4t5.1c1 Dimension 3 Group $S_4$ Conductor $2^{4} \cdot 53$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $848= 2^{4} \cdot 53$ Artin number field: Splitting field of $f= x^{4} - x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Odd Determinant: 1.2e2_53.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 2\cdot 67 + 28\cdot 67^{2} + 27\cdot 67^{3} + 65\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 + 16\cdot 67 + 27\cdot 67^{2} + 37\cdot 67^{3} + 33\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 3 }$ $=$ $58 + 57\cdot 67 + 49\cdot 67^{2} + 54\cdot 67^{3} + 10\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 4 }$ $=$ $61 + 57\cdot 67 + 28\cdot 67^{2} + 14\cdot 67^{3} + 24\cdot 67^{4} +O\left(67^{ 5 }\right)$

### 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.