# Properties

 Label 2.2736.4t3.b Dimension $2$ Group $D_{4}$ Conductor $2736$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$2736$$$$\medspace = 2^{4} \cdot 3^{2} \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.8208.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{-3}, \sqrt{-19})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 8.
Roots:
 $r_{ 1 }$ $=$ $$2 + 2\cdot 7 + 4\cdot 7^{2} + 3\cdot 7^{3} + 4\cdot 7^{4} + 4\cdot 7^{6} + 7^{7} +O(7^{8})$$ $r_{ 2 }$ $=$ $$3 + 5\cdot 7 + 2\cdot 7^{2} + 7^{3} + 4\cdot 7^{4} + 6\cdot 7^{5} + 6\cdot 7^{6} + 2\cdot 7^{7} +O(7^{8})$$ $r_{ 3 }$ $=$ $$4 + 7 + 4\cdot 7^{2} + 5\cdot 7^{3} + 2\cdot 7^{4} + 4\cdot 7^{7} +O(7^{8})$$ $r_{ 4 }$ $=$ $$5 + 4\cdot 7 + 2\cdot 7^{2} + 3\cdot 7^{3} + 2\cdot 7^{4} + 6\cdot 7^{5} + 2\cdot 7^{6} + 5\cdot 7^{7} +O(7^{8})$$

### 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.