# Properties

 Label 1.13.4t1.a Dimension $1$ Group $C_4$ Conductor $13$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$13$$ Artin number field: Galois closure of 4.0.2197.1 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 $\Q_{ 29 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $3 + 20\cdot 29 + 7\cdot 29^{2} + 13\cdot 29^{3} + 11\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 2 }$ $=$ $15 + 25\cdot 29 + 4\cdot 29^{2} + 19\cdot 29^{3} + 6\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 3 }$ $=$ $17 + 28\cdot 29 + 20\cdot 29^{2} + 6\cdot 29^{3} + 6\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 4 }$ $=$ $24 + 12\cdot 29 + 24\cdot 29^{2} + 18\cdot 29^{3} + 4\cdot 29^{4} +O\left(29^{ 5 }\right)$

### 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.