# Properties

 Label 1.52.4t1.a.a Dimension $1$ Group $C_4$ Conductor $52$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$52$$$$\medspace = 2^{2} \cdot 13$$ Artin number field: Galois closure of 4.4.35152.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{52}(31,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{4} - 13 x^{2} + 13$.

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $2 + 8\cdot 23 + 13\cdot 23^{2} + 20\cdot 23^{3} + 4\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $3 + 10\cdot 23 + 17\cdot 23^{2} + 8\cdot 23^{3} + 20\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $20 + 12\cdot 23 + 5\cdot 23^{2} + 14\cdot 23^{3} + 2\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $21 + 14\cdot 23 + 9\cdot 23^{2} + 2\cdot 23^{3} + 18\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,4)(2,3)$ $-1$ $1$ $4$ $(1,2,4,3)$ $\zeta_{4}$ $1$ $4$ $(1,3,4,2)$ $-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.