# Properties

 Label 1.16.4t1.b.a 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: 4.0.2048.2 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{16}(11,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$1 + 4\cdot 7 + 3\cdot 7^{2} + 5\cdot 7^{4} +O(7^{5})$$ $r_{ 2 }$ $=$ $$3 + 3\cdot 7 + 7^{3} + 2\cdot 7^{4} +O(7^{5})$$ $r_{ 3 }$ $=$ $$4 + 3\cdot 7 + 6\cdot 7^{2} + 5\cdot 7^{3} + 4\cdot 7^{4} +O(7^{5})$$ $r_{ 4 }$ $=$ $$6 + 2\cdot 7 + 3\cdot 7^{2} + 6\cdot 7^{3} + 7^{4} +O(7^{5})$$

## 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.