# Properties

 Label 3.15376.4t4.a.a Dimension $3$ Group $A_4$ Conductor $15376$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $$15376$$$$\medspace = 2^{4} \cdot 31^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.0.15376.1 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 4.0.15376.1

## Defining polynomial

 $f(x)$ $=$ $$x^{4} + 7x^{2} - 2x + 14$$ x^4 + 7*x^2 - 2*x + 14 .

The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$60 + 26\cdot 151 + 95\cdot 151^{2} + 134\cdot 151^{3} + 112\cdot 151^{4} +O(151^{5})$$ 60 + 26*151 + 95*151^2 + 134*151^3 + 112*151^4+O(151^5) $r_{ 2 }$ $=$ $$62 + 80\cdot 151 + 36\cdot 151^{2} + 92\cdot 151^{3} + 81\cdot 151^{4} +O(151^{5})$$ 62 + 80*151 + 36*151^2 + 92*151^3 + 81*151^4+O(151^5) $r_{ 3 }$ $=$ $$80 + 24\cdot 151 + 148\cdot 151^{2} + 86\cdot 151^{3} + 107\cdot 151^{4} +O(151^{5})$$ 80 + 24*151 + 148*151^2 + 86*151^3 + 107*151^4+O(151^5) $r_{ 4 }$ $=$ $$100 + 19\cdot 151 + 22\cdot 151^{2} + 139\cdot 151^{3} + 150\cdot 151^{4} +O(151^{5})$$ 100 + 19*151 + 22*151^2 + 139*151^3 + 150*151^4+O(151^5)

## 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.