# Properties

 Label 3.163e2.4t4.1 Dimension 3 Group $A_4$ Conductor $163^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $26569= 163^{2}$ Artin number field: Splitting field of $f= x^{4} - x^{3} - 7 x^{2} + 2 x + 9$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_4$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 241 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $19 + 134\cdot 241 + 70\cdot 241^{2} + 112\cdot 241^{3} + 103\cdot 241^{4} +O\left(241^{ 5 }\right)$ $r_{ 2 }$ $=$ $68 + 209\cdot 241 + 36\cdot 241^{2} + 113\cdot 241^{3} + 113\cdot 241^{4} +O\left(241^{ 5 }\right)$ $r_{ 3 }$ $=$ $196 + 57\cdot 241 + 163\cdot 241^{2} + 122\cdot 241^{3} + 30\cdot 241^{4} +O\left(241^{ 5 }\right)$ $r_{ 4 }$ $=$ $200 + 80\cdot 241 + 211\cdot 241^{2} + 133\cdot 241^{3} + 234\cdot 241^{4} +O\left(241^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,2,3)$ $(1,2)(3,4)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character values $c1$ $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,4)$ $-1$ $4$ $3$ $(1,2,3)$ $0$ $4$ $3$ $(1,3,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.