# Properties

 Label 3.10816.4t4.b.a Dimension $3$ Group $A_4$ Conductor $10816$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $$10816$$$$\medspace = 2^{6} \cdot 13^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.0.10816.1 Galois orbit size: $1$ Smallest permutation container: $A_4$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_4$ Projective stem field: Galois closure of 4.0.10816.1

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 2x^{3} + 2x^{2} + 4x + 2$$ x^4 - 2*x^3 + 2*x^2 + 4*x + 2 .

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$11 + 44\cdot 47 + 23\cdot 47^{2} + 38\cdot 47^{3} + 12\cdot 47^{4} +O(47^{5})$$ 11 + 44*47 + 23*47^2 + 38*47^3 + 12*47^4+O(47^5) $r_{ 2 }$ $=$ $$26 + 33\cdot 47 + 9\cdot 47^{2} + 16\cdot 47^{3} + 10\cdot 47^{4} +O(47^{5})$$ 26 + 33*47 + 9*47^2 + 16*47^3 + 10*47^4+O(47^5) $r_{ 3 }$ $=$ $$29 + 33\cdot 47 + 32\cdot 47^{2} + 30\cdot 47^{3} + 15\cdot 47^{4} +O(47^{5})$$ 29 + 33*47 + 32*47^2 + 30*47^3 + 15*47^4+O(47^5) $r_{ 4 }$ $=$ $$30 + 29\cdot 47 + 27\cdot 47^{2} + 8\cdot 47^{3} + 8\cdot 47^{4} +O(47^{5})$$ 30 + 29*47 + 27*47^2 + 8*47^3 + 8*47^4+O(47^5)

## 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 value $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,4)$ $-1$ $4$ $3$ $(1,2,3)$ $0$ $4$ $3$ $(1,3,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.