# Properties

 Label 3.5_197.4t5.1c1 Dimension 3 Group $S_4$ Conductor $5 \cdot 197$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $985= 5 \cdot 197$ Artin number field: Splitting field of $f=x^{4} - x^{3} + 2 x^{2} - 3 x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Even Determinant: 1.5_197.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 1275252164582 +O\left(311^{ 5 }\right) \\ r_{ 2 } &= 1113684314269 +O\left(311^{ 5 }\right) \\ r_{ 3 } &= -959233306083 +O\left(311^{ 5 }\right) \\ r_{ 4 } &= -1429703172767 +O\left(311^{ 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.