Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(1045\)\(\medspace = 5 \cdot 11 \cdot 19 \) |
| Artin field: | Galois closure of 4.0.5460125.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{1045}(417,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 261x^{2} - 261x + 13781 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 29\cdot 41 + 5\cdot 41^{2} + 4\cdot 41^{3} + 28\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 14\cdot 41 + 34\cdot 41^{2} + 13\cdot 41^{3} + 2\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 30\cdot 41 + 15\cdot 41^{2} + 26\cdot 41^{3} + 25\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 7\cdot 41 + 26\cdot 41^{2} + 37\cdot 41^{3} + 25\cdot 41^{4} +O(41^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.