# Properties

 Label 3.121801.4t4.a.a Dimension $3$ Group $A_4$ Conductor $121801$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $$121801$$$$\medspace = 349^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.4.121801.1 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 4.4.121801.1

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} - 10x^{2} + 3x + 20$$ x^4 - x^3 - 10*x^2 + 3*x + 20 .

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$9 + 5\cdot 37 + 27\cdot 37^{2} + 10\cdot 37^{3} +O(37^{5})$$ 9 + 5*37 + 27*37^2 + 10*37^3+O(37^5) $r_{ 2 }$ $=$ $$18 + 28\cdot 37 + 22\cdot 37^{2} + 31\cdot 37^{3} + 17\cdot 37^{4} +O(37^{5})$$ 18 + 28*37 + 22*37^2 + 31*37^3 + 17*37^4+O(37^5) $r_{ 3 }$ $=$ $$21 + 30\cdot 37 + 8\cdot 37^{2} + 7\cdot 37^{3} + 12\cdot 37^{4} +O(37^{5})$$ 21 + 30*37 + 8*37^2 + 7*37^3 + 12*37^4+O(37^5) $r_{ 4 }$ $=$ $$27 + 9\cdot 37 + 15\cdot 37^{2} + 24\cdot 37^{3} + 6\cdot 37^{4} +O(37^{5})$$ 27 + 9*37 + 15*37^2 + 24*37^3 + 6*37^4+O(37^5)

## 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.