Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(101\) |
| Artin field: | Galois closure of 4.0.1030301.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{101}(91,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 13x^{2} - 19x + 361 \)
|
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 30\cdot 31 + 7\cdot 31^{2} + 15\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 29\cdot 31 + 12\cdot 31^{2} + 18\cdot 31^{3} + 15\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 4\cdot 31 + 12\cdot 31^{2} + 9\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 28\cdot 31 + 28\cdot 31^{2} + 18\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.