# Properties

 Label 3.1207801.6t8.a Dimension $3$ Group $S_4$ Conductor $1207801$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $$1207801$$$$\medspace = 7^{2} \cdot 157^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.2.1099.1 Galois orbit size: $1$ Smallest permutation container: $S_4$ Parity: even Projective image: $S_4$ Projective field: Galois closure of 4.2.1099.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 317 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $116 + 117\cdot 317 + 120\cdot 317^{2} + 83\cdot 317^{3} + 188\cdot 317^{4} +O\left(317^{ 5 }\right)$ $r_{ 2 }$ $=$ $168 + 199\cdot 317 + 307\cdot 317^{2} + 80\cdot 317^{3} + 173\cdot 317^{4} +O\left(317^{ 5 }\right)$ $r_{ 3 }$ $=$ $170 + 64\cdot 317 + 126\cdot 317^{2} + 41\cdot 317^{3} + 56\cdot 317^{4} +O\left(317^{ 5 }\right)$ $r_{ 4 }$ $=$ $181 + 252\cdot 317 + 79\cdot 317^{2} + 111\cdot 317^{3} + 216\cdot 317^{4} +O\left(317^{ 5 }\right)$

### 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 values $c1$ $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,4)$ $-1$ $6$ $2$ $(1,2)$ $-1$ $8$ $3$ $(1,2,3)$ $0$ $6$ $4$ $(1,2,3,4)$ $1$
The blue line marks the conjugacy class containing complex conjugation.