# Properties

 Label 2.2e3_7.4t3.2c1 Dimension 2 Group $D_{4}$ Conductor $2^{3} \cdot 7$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $56= 2^{3} \cdot 7$ Artin number field: Splitting field of $f= x^{4} - x^{3} + x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.2e3_7.2t1.2c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $5 + 2\cdot 23 + 17\cdot 23^{2} + 22\cdot 23^{3} + 21\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $9 + 19\cdot 23 + 16\cdot 23^{2} + 2\cdot 23^{3} + 7\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $12 + 9\cdot 23 + 13\cdot 23^{2} + 12\cdot 23^{3} + 6\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $21 + 14\cdot 23 + 21\cdot 23^{2} + 7\cdot 23^{3} + 10\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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