# Properties

 Label 3.1366561.4t4.a Dimension $3$ Group $A_4$ Conductor $1366561$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $$1366561$$$$\medspace = 7^{2} \cdot 167^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.4.1366561.1 Galois orbit size: $1$ Smallest permutation container: $A_4$ Parity: even Projective image: $A_4$ Projective field: Galois closure of 4.4.1366561.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$11 + 33\cdot 43 + 13\cdot 43^{2} + 24\cdot 43^{3} + 14\cdot 43^{4} +O(43^{5})$$ 11 + 33*43 + 13*43^2 + 24*43^3 + 14*43^4+O(43^5) $r_{ 2 }$ $=$ $$20 + 31\cdot 43 + 11\cdot 43^{2} + 27\cdot 43^{3} + 20\cdot 43^{4} +O(43^{5})$$ 20 + 31*43 + 11*43^2 + 27*43^3 + 20*43^4+O(43^5) $r_{ 3 }$ $=$ $$24 + 7\cdot 43 + 34\cdot 43^{2} + 23\cdot 43^{3} + 22\cdot 43^{4} +O(43^{5})$$ 24 + 7*43 + 34*43^2 + 23*43^3 + 22*43^4+O(43^5) $r_{ 4 }$ $=$ $$32 + 13\cdot 43 + 26\cdot 43^{2} + 10\cdot 43^{3} + 28\cdot 43^{4} +O(43^{5})$$ 32 + 13*43 + 26*43^2 + 10*43^3 + 28*43^4+O(43^5)

### 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.