# Properties

 Label 1.5.4t1.a Dimension $1$ Group $C_4$ Conductor $5$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$5$$ Artin number field: Galois closure of $$\Q(\zeta_{5})$$ Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$2 + 10\cdot 11 + 4\cdot 11^{2} + 9\cdot 11^{3} + 11^{4} +O(11^{5})$$ $r_{ 2 }$ $=$ $$6 + 8\cdot 11 + 5\cdot 11^{2} + 9\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})$$ $r_{ 3 }$ $=$ $$7 + 3\cdot 11 + 11^{2} + 5\cdot 11^{3} + 8\cdot 11^{4} +O(11^{5})$$ $r_{ 4 }$ $=$ $$8 + 10\cdot 11 + 9\cdot 11^{2} + 8\cdot 11^{3} + 7\cdot 11^{4} +O(11^{5})$$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,2)(3,4)$ $-1$ $-1$ $1$ $4$ $(1,4,2,3)$ $\zeta_{4}$ $-\zeta_{4}$ $1$ $4$ $(1,3,2,4)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.