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
| $f(x)$ | $=$ |
\( x^{4} + 12x^{2} + 18 \)
|
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 8\cdot 23 + 8\cdot 23^{2} + 13\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 18\cdot 23 + 6\cdot 23^{2} + 20\cdot 23^{3} + 2\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 4\cdot 23 + 16\cdot 23^{2} + 2\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 14\cdot 23 + 14\cdot 23^{2} + 9\cdot 23^{3} + 2\cdot 23^{4} +O(23^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ | ✓ |
| $1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ | |
| $1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ |