# Properties

 Label 2.256.4t3.c.a Dimension 2 Group $D_{4}$ Conductor $2^{8}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $256= 2^{8}$ Artin number field: Splitting field of 4.2.2048.1 defined by $f= x^{4} - 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.4.2t1.a.a 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 $\Q_{ 73 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $18 + 13\cdot 73 + 54\cdot 73^{2} + 73^{3} + 26\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 2 }$ $=$ $25 + 59\cdot 73 + 58\cdot 73^{2} + 32\cdot 73^{3} + 47\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 3 }$ $=$ $48 + 13\cdot 73 + 14\cdot 73^{2} + 40\cdot 73^{3} + 25\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 4 }$ $=$ $55 + 59\cdot 73 + 18\cdot 73^{2} + 71\cdot 73^{3} + 46\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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