Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(1036\)\(\medspace = 2^{2} \cdot 7 \cdot 37 \) |
| Artin field: | Galois closure of 4.0.39711952.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{1036}(475,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 259x^{2} + 16317 \)
|
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 10\cdot 11^{2} + 6\cdot 11^{3} + 3\cdot 11^{5} + 7\cdot 11^{6} +O(11^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 5\cdot 11 + 7\cdot 11^{2} + 8\cdot 11^{3} + 6\cdot 11^{4} + 2\cdot 11^{5} + 4\cdot 11^{6} + 4\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 5\cdot 11 + 3\cdot 11^{2} + 2\cdot 11^{3} + 4\cdot 11^{4} + 8\cdot 11^{5} + 6\cdot 11^{6} + 6\cdot 11^{7} +O(11^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 + 10\cdot 11 + 4\cdot 11^{3} + 10\cdot 11^{4} + 7\cdot 11^{5} + 3\cdot 11^{6} + 10\cdot 11^{7} +O(11^{8})\)
|
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 value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.