# Properties

 Label 1.105.4t1.a Dimension $1$ Group $C_4$ Conductor $105$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$105$$$$\medspace = 3 \cdot 5 \cdot 7$$ Artin number field: Galois closure of 4.0.55125.1 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 $\Q_{ 19 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$3 + 9\cdot 19^{2} + 7\cdot 19^{3} + 9\cdot 19^{4} +O(19^{5})$$ $r_{ 2 }$ $=$ $$11 + 14\cdot 19 + 18\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})$$ $r_{ 3 }$ $=$ $$12 + 16\cdot 19 + 18\cdot 19^{2} + 15\cdot 19^{3} + 19^{4} +O(19^{5})$$ $r_{ 4 }$ $=$ $$13 + 6\cdot 19 + 9\cdot 19^{2} + 15\cdot 19^{3} + 13\cdot 19^{4} +O(19^{5})$$

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

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

### Character values on conjugacy classes

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