# Properties

 Label 2.768.4t3.c 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 4.0.6144.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-2}, \sqrt{-3})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$7 + 60\cdot 67 + 41\cdot 67^{2} + 28\cdot 67^{3} + 47\cdot 67^{4} +O(67^{5})$$ 7 + 60*67 + 41*67^2 + 28*67^3 + 47*67^4+O(67^5) $r_{ 2 }$ $=$ $$25 + 50\cdot 67 + 29\cdot 67^{2} + 37\cdot 67^{3} + 28\cdot 67^{4} +O(67^{5})$$ 25 + 50*67 + 29*67^2 + 37*67^3 + 28*67^4+O(67^5) $r_{ 3 }$ $=$ $$42 + 16\cdot 67 + 37\cdot 67^{2} + 29\cdot 67^{3} + 38\cdot 67^{4} +O(67^{5})$$ 42 + 16*67 + 37*67^2 + 29*67^3 + 38*67^4+O(67^5) $r_{ 4 }$ $=$ $$60 + 6\cdot 67 + 25\cdot 67^{2} + 38\cdot 67^{3} + 19\cdot 67^{4} +O(67^{5})$$ 60 + 6*67 + 25*67^2 + 38*67^3 + 19*67^4+O(67^5)

### 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.