# Properties

 Label 2.219.4t3.c.a Dimension 2 Group $D_{4}$ Conductor $3 \cdot 73$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $219= 3 \cdot 73$ Artin number field: Splitting field of 4.0.657.1 defined by $f= x^{4} - x^{3} + 5 x^{2} - 2 x + 4$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.219.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{73})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $19 + 34\cdot 61 + 60\cdot 61^{2} + 3\cdot 61^{3} + 30\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 + 16\cdot 61 + 41\cdot 61^{2} + 46\cdot 61^{3} + 38\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 3 }$ $=$ $29 + 60\cdot 61 + 40\cdot 61^{2} + 29\cdot 61^{3} + 57\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 4 }$ $=$ $55 + 10\cdot 61 + 40\cdot 61^{2} + 41\cdot 61^{3} + 56\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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