# Properties

 Label 2.768.4t3.d Dimension $2$ Group $D_{4}$ Conductor $768$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$768$$$$\medspace = 2^{8} \cdot 3$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.6144.2 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{-2}, \sqrt{-3})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$2 + 15\cdot 19 + 16\cdot 19^{2} + 15\cdot 19^{3} + 7\cdot 19^{4} +O(19^{5})$$ $r_{ 2 }$ $=$ $$7 + 5\cdot 19 + 11\cdot 19^{2} + 2\cdot 19^{3} +O(19^{5})$$ $r_{ 3 }$ $=$ $$12 + 13\cdot 19 + 7\cdot 19^{2} + 16\cdot 19^{3} + 18\cdot 19^{4} +O(19^{5})$$ $r_{ 4 }$ $=$ $$17 + 3\cdot 19 + 2\cdot 19^{2} + 3\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})$$

### 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 values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,4)$ $0$ $2$ $4$ $(1,3,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.