# Properties

 Label 3.3e3_41.4t5.1c1 Dimension 3 Group $S_4$ Conductor $3^{3} \cdot 41$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $1107= 3^{3} \cdot 41$ Artin number field: Splitting field of $f= x^{4} - x^{3} - 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Odd Determinant: 1.3_41.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 317 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $67 + 168\cdot 317 + 121\cdot 317^{2} + 285\cdot 317^{3} + 274\cdot 317^{4} +O\left(317^{ 5 }\right)$ $r_{ 2 }$ $=$ $126 + 290\cdot 317 + 149\cdot 317^{2} + 25\cdot 317^{3} + 111\cdot 317^{4} +O\left(317^{ 5 }\right)$ $r_{ 3 }$ $=$ $205 + 163\cdot 317 + 45\cdot 317^{2} + 92\cdot 317^{3} + 103\cdot 317^{4} +O\left(317^{ 5 }\right)$ $r_{ 4 }$ $=$ $237 + 11\cdot 317 + 231\cdot 317^{3} + 144\cdot 317^{4} +O\left(317^{ 5 }\right)$

### 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 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.