# Properties

 Label 3.3e4_37e2.4t4.1 Dimension 3 Group $A_4$ Conductor $3^{4} \cdot 37^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $110889= 3^{4} \cdot 37^{2}$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 6 x^{2} + 11 x + 10$ 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 $\Q_{ 19 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $4 + 19 + 9\cdot 19^{2} + 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $8 + 15\cdot 19 + 2\cdot 19^{2} + 19^{3} + 16\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $13 + 17\cdot 19 + 19^{2} + 16\cdot 19^{3} + 5\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $14 + 3\cdot 19 + 5\cdot 19^{2} + 19^{3} + 15\cdot 19^{4} +O\left(19^{ 5 }\right)$

### 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.