# Properties

 Label 3.29241.4t4.b Dimension $3$ Group $A_4$ Conductor $29241$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $$29241$$$$\medspace = 3^{4} \cdot 19^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.29241.1 Galois orbit size: $1$ Smallest permutation container: $A_4$ Parity: even Projective image: $A_4$ Projective field: Galois closure of 4.0.29241.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$9 + 54\cdot 109 + 74\cdot 109^{2} + 48\cdot 109^{3} + 50\cdot 109^{4} +O(109^{5})$$ 9 + 54*109 + 74*109^2 + 48*109^3 + 50*109^4+O(109^5) $r_{ 2 }$ $=$ $$15 + 8\cdot 109 + 51\cdot 109^{2} + 45\cdot 109^{3} + 45\cdot 109^{4} +O(109^{5})$$ 15 + 8*109 + 51*109^2 + 45*109^3 + 45*109^4+O(109^5) $r_{ 3 }$ $=$ $$32 + 15\cdot 109 + 82\cdot 109^{2} + 48\cdot 109^{3} + 58\cdot 109^{4} +O(109^{5})$$ 32 + 15*109 + 82*109^2 + 48*109^3 + 58*109^4+O(109^5) $r_{ 4 }$ $=$ $$54 + 31\cdot 109 + 10\cdot 109^{2} + 75\cdot 109^{3} + 63\cdot 109^{4} +O(109^{5})$$ 54 + 31*109 + 10*109^2 + 75*109^3 + 63*109^4+O(109^5)

### 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.