Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Artin field: | Galois closure of 4.0.8000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{40}(37,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 10x^{2} + 20 \)
|
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 19 + 12\cdot 19^{3} + 2\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 9\cdot 19 + 15\cdot 19^{2} + 14\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 9\cdot 19 + 3\cdot 19^{2} + 4\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 17\cdot 19 + 18\cdot 19^{2} + 6\cdot 19^{3} + 16\cdot 19^{4} +O(19^{5})\)
|
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.