# Properties

 Label 1.13.4t1.a.a Dimension $1$ Group $C_4$ Conductor $13$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$13$$ Artin field: 4.0.2197.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{13}(8,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} + 2 x^{2} + 4 x + 3$$  .

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$3 + 20\cdot 29 + 7\cdot 29^{2} + 13\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})$$ $r_{ 2 }$ $=$ $$15 + 25\cdot 29 + 4\cdot 29^{2} + 19\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})$$ $r_{ 3 }$ $=$ $$17 + 28\cdot 29 + 20\cdot 29^{2} + 6\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})$$ $r_{ 4 }$ $=$ $$24 + 12\cdot 29 + 24\cdot 29^{2} + 18\cdot 29^{3} + 4\cdot 29^{4} +O(29^{5})$$

## 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 value $1$ $1$ $()$ $1$ $1$ $2$ $(1,3)(2,4)$ $-1$ $1$ $4$ $(1,2,3,4)$ $\zeta_{4}$ $1$ $4$ $(1,4,3,2)$ $-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.