# Properties

 Label 1.16.4t1.a.b Dimension $1$ Group $C_4$ Conductor $16$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$16$$$$\medspace = 2^{4}$$ Artin field: Galois closure of $$\Q(\zeta_{16})^+$$ Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{16}(13,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$5 + 3\cdot 17 + 14\cdot 17^{2} + 12\cdot 17^{3} + 15\cdot 17^{4} +O(17^{5})$$ 5 + 3*17 + 14*17^2 + 12*17^3 + 15*17^4+O(17^5) $r_{ 2 }$ $=$ $$8 + 17 + 9\cdot 17^{3} + 15\cdot 17^{4} +O(17^{5})$$ 8 + 17 + 9*17^3 + 15*17^4+O(17^5) $r_{ 3 }$ $=$ $$9 + 15\cdot 17 + 16\cdot 17^{2} + 7\cdot 17^{3} + 17^{4} +O(17^{5})$$ 9 + 15*17 + 16*17^2 + 7*17^3 + 17^4+O(17^5) $r_{ 4 }$ $=$ $$12 + 13\cdot 17 + 2\cdot 17^{2} + 4\cdot 17^{3} + 17^{4} +O(17^{5})$$ 12 + 13*17 + 2*17^2 + 4*17^3 + 17^4+O(17^5)

## 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.