# Properties

 Label 1.16.4t1.a.a Dimension 1 Group $C_4$ Conductor $2^{4}$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $16= 2^{4}$ Artin number field: Splitting field of $$\Q(\zeta_{16})^+$$ defined by $f= x^{4} - 4 x^{2} + 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_4$ Parity: Even Corresponding Dirichlet character: $$\chi_{16}(5,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $5 + 3\cdot 17 + 14\cdot 17^{2} + 12\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 2 }$ $=$ $8 + 17 + 9\cdot 17^{3} + 15\cdot 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 3 }$ $=$ $9 + 15\cdot 17 + 16\cdot 17^{2} + 7\cdot 17^{3} + 17^{4} +O\left(17^{ 5 }\right)$ $r_{ 4 }$ $=$ $12 + 13\cdot 17 + 2\cdot 17^{2} + 4\cdot 17^{3} + 17^{4} +O\left(17^{ 5 }\right)$

### 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.