# Properties

 Label 3.2e6_53e2.6t8.3c1 Dimension 3 Group $S_4$ Conductor $2^{6} \cdot 53^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $179776= 2^{6} \cdot 53^{2}$ 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: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots: \begin{aligned} 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) \\ \end{aligned}

### 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.