# Properties

 Label 2.56.4t3.b Dimension $2$ Group $D_{4}$ Conductor $56$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$56$$$$\medspace = 2^{3} \cdot 7$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.392.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{2}, \sqrt{-7})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$5 + 2\cdot 23 + 17\cdot 23^{2} + 22\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})$$ 5 + 2*23 + 17*23^2 + 22*23^3 + 21*23^4+O(23^5) $r_{ 2 }$ $=$ $$9 + 19\cdot 23 + 16\cdot 23^{2} + 2\cdot 23^{3} + 7\cdot 23^{4} +O(23^{5})$$ 9 + 19*23 + 16*23^2 + 2*23^3 + 7*23^4+O(23^5) $r_{ 3 }$ $=$ $$12 + 9\cdot 23 + 13\cdot 23^{2} + 12\cdot 23^{3} + 6\cdot 23^{4} +O(23^{5})$$ 12 + 9*23 + 13*23^2 + 12*23^3 + 6*23^4+O(23^5) $r_{ 4 }$ $=$ $$21 + 14\cdot 23 + 21\cdot 23^{2} + 7\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})$$ 21 + 14*23 + 21*23^2 + 7*23^3 + 10*23^4+O(23^5)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,2)(3,4)$ $-2$ $2$ $2$ $(1,3)(2,4)$ $0$ $2$ $2$ $(1,2)$ $0$ $2$ $4$ $(1,4,2,3)$ $0$
The blue line marks the conjugacy class containing complex conjugation.