# Properties

 Label 3.2e4_89.4t5.1c1 Dimension 3 Group $S_4$ Conductor $2^{4} \cdot 89$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $1424= 2^{4} \cdot 89$ Artin number field: Splitting field of $f= x^{4} + x^{2} - 2 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Odd Determinant: 1.2e2_89.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 20\cdot 83 + 59\cdot 83^{2} + 64\cdot 83^{3} + 75\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 2 }$ $=$ $32 + 74\cdot 83 + 19\cdot 83^{2} + 23\cdot 83^{3} + 70\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 3 }$ $=$ $63 + 5\cdot 83 + 16\cdot 83^{2} + 15\cdot 83^{3} + 30\cdot 83^{4} +O\left(83^{ 5 }\right)$ $r_{ 4 }$ $=$ $68 + 65\cdot 83 + 70\cdot 83^{2} + 62\cdot 83^{3} + 72\cdot 83^{4} +O\left(83^{ 5 }\right)$

### 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.