# Properties

 Label 3.277e2.4t4.1c1 Dimension 3 Group $A_4$ Conductor $277^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $76729= 277^{2}$ Artin number field: Splitting field of $f= x^{4} - x^{3} - 11 x^{2} + 4 x + 12$ 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 $\Q_{ 149 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $47 + 31\cdot 149 + 124\cdot 149^{2} + 23\cdot 149^{3} + 115\cdot 149^{4} +O\left(149^{ 5 }\right)$ $r_{ 2 }$ $=$ $48 + 54\cdot 149 + 147\cdot 149^{2} + 99\cdot 149^{3} + 56\cdot 149^{4} +O\left(149^{ 5 }\right)$ $r_{ 3 }$ $=$ $98 + 109\cdot 149 + 41\cdot 149^{2} + 62\cdot 149^{3} + 117\cdot 149^{4} +O\left(149^{ 5 }\right)$ $r_{ 4 }$ $=$ $106 + 102\cdot 149 + 133\cdot 149^{2} + 111\cdot 149^{3} + 8\cdot 149^{4} +O\left(149^{ 5 }\right)$

### 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.