# Properties

 Label 3.397e2.4t4.1c1 Dimension 3 Group $A_4$ Conductor $397^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $157609= 397^{2}$ Artin number field: Splitting field of $f= x^{4} - 13 x^{2} - 2 x + 19$ 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 $\Q_{ 103 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $5 + 82\cdot 103 + 8\cdot 103^{2} + 21\cdot 103^{3} + 31\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 2 }$ $=$ $15 + 21\cdot 103 + 6\cdot 103^{2} + 53\cdot 103^{3} + 78\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 3 }$ $=$ $22 + 77\cdot 103 + 2\cdot 103^{2} + 19\cdot 103^{3} + 85\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 4 }$ $=$ $61 + 25\cdot 103 + 85\cdot 103^{2} + 9\cdot 103^{3} + 11\cdot 103^{4} +O\left(103^{ 5 }\right)$

### 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.