# Properties

 Label 2.295.4t3.c Dimension 2 Group $D_{4}$ Conductor $5 \cdot 59$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $295= 5 \cdot 59$ Artin number field: Splitting field of $f= x^{4} - x^{3} - x^{2} + 5 x - 5$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-59})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $8 + 52\cdot 79 + 39\cdot 79^{2} + 67\cdot 79^{3} + 43\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 2 }$ $=$ $33 + 51\cdot 79 + 68\cdot 79^{2} + 46\cdot 79^{3} + 78\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 3 }$ $=$ $42 + 61\cdot 79 + 62\cdot 79^{2} + 57\cdot 79^{3} + 26\cdot 79^{4} +O\left(79^{ 5 }\right)$ $r_{ 4 }$ $=$ $76 + 71\cdot 79 + 65\cdot 79^{2} + 64\cdot 79^{3} + 8\cdot 79^{4} +O\left(79^{ 5 }\right)$

### 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 values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,3)(2,4)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,3)$ $0$ $2$ $4$ $(1,4,3,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.