# Properties

 Label 3.7e2_71e2.4t4.1 Dimension 3 Group $A_4$ Conductor $7^{2} \cdot 71^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $247009= 7^{2} \cdot 71^{2}$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 9 x + 10$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $A_4$ Parity: Even

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 7\cdot 13 + 5\cdot 13^{2} + 7\cdot 13^{3} + 4\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 + 5\cdot 13 + 6\cdot 13^{2} + 8\cdot 13^{3} +O\left(13^{ 5 }\right)$ $r_{ 3 }$ $=$ $8 + 5\cdot 13 + 4\cdot 13^{2} + 13^{3} + 12\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 4 }$ $=$ $10 + 7\cdot 13 + 9\cdot 13^{2} + 8\cdot 13^{3} + 8\cdot 13^{4} +O\left(13^{ 5 }\right)$

### 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 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.