Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(29\) |
| Artin field: | Galois closure of 4.0.24389.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{29}(12,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 4x^{2} - 20x + 23 \)
|
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15\cdot 23 + 11\cdot 23^{2} + 16\cdot 23^{3} + 5\cdot 23^{4} +O(23^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 6 + 10\cdot 23 + 21\cdot 23^{2} + 4\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 8 + 11\cdot 23^{2} + 9\cdot 23^{3} +O(23^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 10 + 20\cdot 23 + 23^{2} + 15\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.