# Properties

 Label 3.3e2_103e2.4t4.1c1 Dimension 3 Group $A_4$ Conductor $3^{2} \cdot 103^{2}$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $95481= 3^{2} \cdot 103^{2}$ Artin number field: Splitting field of $f= x^{4} + 13 x^{2} - 3 x + 46$ 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 $\Q_{ 23 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $16\cdot 23 + 20\cdot 23^{2} + 7\cdot 23^{3} + 8\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $10 + 7\cdot 23 + 9\cdot 23^{2} + 7\cdot 23^{3} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $16 + 6\cdot 23 + 23^{2} + 20\cdot 23^{3} + 12\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $20 + 15\cdot 23 + 14\cdot 23^{2} + 10\cdot 23^{3} + 23^{4} +O\left(23^{ 5 }\right)$

### 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.