# Properties

 Label 2.4645.4t3.c Dimension $2$ Group $D_{4}$ Conductor $4645$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$4645$$$$\medspace = 5 \cdot 929$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.4315205.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: even Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{929})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $7 + 26\cdot 29 + 10\cdot 29^{2} + 22\cdot 29^{3} + 27\cdot 29^{4} + 24\cdot 29^{5} +O\left(29^{ 6 }\right)$ $r_{ 2 }$ $=$ $9 + 2\cdot 29 + 13\cdot 29^{2} + 2\cdot 29^{3} + 15\cdot 29^{4} + 4\cdot 29^{5} +O\left(29^{ 6 }\right)$ $r_{ 3 }$ $=$ $21 + 26\cdot 29 + 15\cdot 29^{2} + 26\cdot 29^{3} + 13\cdot 29^{4} + 24\cdot 29^{5} +O\left(29^{ 6 }\right)$ $r_{ 4 }$ $=$ $23 + 2\cdot 29 + 18\cdot 29^{2} + 6\cdot 29^{3} + 29^{4} + 4\cdot 29^{5} +O\left(29^{ 6 }\right)$

### 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.