# Properties

 Label 2.128.4t3.a Dimension 2 Group $D_{4}$ Conductor $2^{7}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $128= 2^{7}$ Artin number field: Splitting field of $f= x^{4} - 2 x^{2} + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\zeta_{8})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $16 + 27\cdot 41 + 34\cdot 41^{2} + 6\cdot 41^{3} + 17\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 + 6\cdot 41 + 39\cdot 41^{2} + 12\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $22 + 34\cdot 41 + 41^{2} + 28\cdot 41^{3} + 2\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $25 + 13\cdot 41 + 6\cdot 41^{2} + 34\cdot 41^{3} + 23\cdot 41^{4} +O\left(41^{ 5 }\right)$

### 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.