# Properties

 Label 3.3e4_5e2_7e2.4t4.1c1 Dimension 3 Group $A_4$ Conductor $3^{4} \cdot 5^{2} \cdot 7^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $99225= 3^{4} \cdot 5^{2} \cdot 7^{2}$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 7 x + 14$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_4$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $6 + 64\cdot 71 + 25\cdot 71^{2} + 38\cdot 71^{3} + 65\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 2 }$ $=$ $14 + 12\cdot 71 + 25\cdot 71^{2} + 49\cdot 71^{3} + 50\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 3 }$ $=$ $21 + 20\cdot 71 + 55\cdot 71^{2} + 26\cdot 71^{3} + 38\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 4 }$ $=$ $31 + 45\cdot 71 + 35\cdot 71^{2} + 27\cdot 71^{3} + 58\cdot 71^{4} +O\left(71^{ 5 }\right)$

### 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.