# Properties

 Label 3.7344.4t5.a.a Dimension 3 Group $S_4$ Conductor $2^{4} \cdot 3^{3} \cdot 17$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $7344= 2^{4} \cdot 3^{3} \cdot 17$ Artin number field: Splitting field of 4.2.7344.1 defined by $f= x^{4} - 2 x^{3} - 2 x - 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Odd Determinant: 1.51.2t1.a.a Projective image: $S_4$ Projective field: Galois closure of 4.2.7344.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $21 + 60\cdot 151 + 137\cdot 151^{2} + 90\cdot 151^{3} + 81\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 2 }$ $=$ $51 + 35\cdot 151 + 116\cdot 151^{2} + 40\cdot 151^{3} +O\left(151^{ 5 }\right)$ $r_{ 3 }$ $=$ $89 + 137\cdot 151 + 135\cdot 151^{2} + 90\cdot 151^{3} + 104\cdot 151^{4} +O\left(151^{ 5 }\right)$ $r_{ 4 }$ $=$ $143 + 68\cdot 151 + 63\cdot 151^{2} + 79\cdot 151^{3} + 115\cdot 151^{4} +O\left(151^{ 5 }\right)$

### 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.