Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(16\)\(\medspace = 2^{4} \) |
| Artin field: | Galois closure of 4.0.2048.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{16}(3,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 4x^{2} + 2 \)
|
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 |
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.