# Properties

 Label 2.384.4t3.a Dimension $2$ Group $D_{4}$ Conductor $384$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$384$$$$\medspace = 2^{7} \cdot 3$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.3072.1 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 $\Q_{ 11 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $2 + 5\cdot 11 + 8\cdot 11^{3} + 5\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 + 2\cdot 11 + 2\cdot 11^{2} + 4\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 3 }$ $=$ $8 + 8\cdot 11 + 8\cdot 11^{2} + 10\cdot 11^{3} + 6\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 4 }$ $=$ $9 + 5\cdot 11 + 10\cdot 11^{2} + 2\cdot 11^{3} + 5\cdot 11^{4} +O\left(11^{ 5 }\right)$

### 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.