# Properties

 Label 2.63.4t3.a.a Dimension 2 Group $D_{4}$ Conductor $3^{2} \cdot 7$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $63= 3^{2} \cdot 7$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.7.2t1.a.a

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $4 + 22\cdot 67 + 31\cdot 67^{2} + 41\cdot 67^{3} + 49\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 2 }$ $=$ $11 + 54\cdot 67 + 30\cdot 67^{2} + 61\cdot 67^{3} + 5\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 3 }$ $=$ $26 + 55\cdot 67 + 57\cdot 67^{2} + 24\cdot 67^{3} + 59\cdot 67^{4} +O\left(67^{ 5 }\right)$ $r_{ 4 }$ $=$ $27 + 2\cdot 67 + 14\cdot 67^{2} + 6\cdot 67^{3} + 19\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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