# Properties

 Label 2.2e4_3e2.4t3.2c1 Dimension 2 Group $D_{4}$ Conductor $2^{4} \cdot 3^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $144= 2^{4} \cdot 3^{2}$ Artin number field: Splitting field of $f= x^{4} - 3 x^{2} + 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.2e2.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $7 + 12\cdot 37 + 23\cdot 37^{2} + 25\cdot 37^{3} + 26\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $18 + 30\cdot 37 + 31\cdot 37^{2} + 18\cdot 37^{3} + 2\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $19 + 6\cdot 37 + 5\cdot 37^{2} + 18\cdot 37^{3} + 34\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $30 + 24\cdot 37 + 13\cdot 37^{2} + 11\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,4)$ $(1,2)(3,4)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,4)$ $0$ $2$ $4$ $(1,3,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.