# Properties

 Label 2.7_11_43.4t3.4 Dimension 2 Group $D_{4}$ Conductor $7 \cdot 11 \cdot 43$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $3311= 7 \cdot 11 \cdot 43$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 11 x^{2} - 10 x + 32$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $9 + 30\cdot 53 + 11\cdot 53^{2} + 20\cdot 53^{3} + 8\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 + 39\cdot 53 + 3\cdot 53^{2} + 34\cdot 53^{3} + 2\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 3 }$ $=$ $35 + 13\cdot 53 + 49\cdot 53^{2} + 18\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 4 }$ $=$ $45 + 22\cdot 53 + 41\cdot 53^{2} + 32\cdot 53^{3} + 44\cdot 53^{4} +O\left(53^{ 5 }\right)$

### 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 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.