# Properties

 Label 3.2e4_7_13.4t5.1c1 Dimension 3 Group $S_4$ Conductor $2^{4} \cdot 7 \cdot 13$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $1456= 2^{4} \cdot 7 \cdot 13$ Artin number field: Splitting field of $f= x^{4} - 2 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Odd Determinant: 1.7_13.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $8 + 58\cdot 59 + 7\cdot 59^{2} + 3\cdot 59^{3} + 25\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 2 }$ $=$ $24 + 9\cdot 59 + 21\cdot 59^{2} + 54\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 3 }$ $=$ $39 + 53\cdot 59 + 33\cdot 59^{2} + 34\cdot 59^{3} + 8\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 4 }$ $=$ $47 + 55\cdot 59 + 54\cdot 59^{2} + 20\cdot 59^{3} + 30\cdot 59^{4} +O\left(59^{ 5 }\right)$

### 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.