# Properties

 Label 1.48.4t1.b.a Dimension $1$ Group $C_4$ Conductor $48$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$48$$$$\medspace = 2^{4} \cdot 3$$ Artin field: Galois closure of 4.0.18432.2 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{48}(5,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} + 12x^{2} + 18$$ x^4 + 12*x^2 + 18 .

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

Roots:
 $r_{ 1 }$ $=$ $$3 + 8\cdot 23 + 8\cdot 23^{2} + 13\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})$$ 3 + 8*23 + 8*23^2 + 13*23^3 + 20*23^4+O(23^5) $r_{ 2 }$ $=$ $$5 + 18\cdot 23 + 6\cdot 23^{2} + 20\cdot 23^{3} + 2\cdot 23^{4} +O(23^{5})$$ 5 + 18*23 + 6*23^2 + 20*23^3 + 2*23^4+O(23^5) $r_{ 3 }$ $=$ $$18 + 4\cdot 23 + 16\cdot 23^{2} + 2\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})$$ 18 + 4*23 + 16*23^2 + 2*23^3 + 20*23^4+O(23^5) $r_{ 4 }$ $=$ $$20 + 14\cdot 23 + 14\cdot 23^{2} + 9\cdot 23^{3} + 2\cdot 23^{4} +O(23^{5})$$ 20 + 14*23 + 14*23^2 + 9*23^3 + 2*23^4+O(23^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.