Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(10064\)\(\medspace = 2^{4} \cdot 17 \cdot 37 \) |
| Artin number field: | Galois closure of 4.0.810272768.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 13\cdot 23 + 5\cdot 23^{2} + 9\cdot 23^{3} + 2\cdot 23^{4} + 4\cdot 23^{5} + 7\cdot 23^{6} + 15\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 2\cdot 23 + 11\cdot 23^{2} + 2\cdot 23^{3} + 23^{4} + 5\cdot 23^{5} + 21\cdot 23^{6} + 17\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 20\cdot 23 + 11\cdot 23^{2} + 20\cdot 23^{3} + 21\cdot 23^{4} + 17\cdot 23^{5} + 23^{6} + 5\cdot 23^{7} +O(23^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 9\cdot 23 + 17\cdot 23^{2} + 13\cdot 23^{3} + 20\cdot 23^{4} + 18\cdot 23^{5} + 15\cdot 23^{6} + 7\cdot 23^{7} +O(23^{8})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ | $-1$ |
| $1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ | $\zeta_{4}$ |