Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(143\)\(\medspace = 11 \cdot 13 \) |
| Artin field: | Galois closure of 4.4.265837.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{143}(109,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 37x^{2} - 35x + 81 \)
|
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 29\cdot 43 + 39\cdot 43^{2} + 35\cdot 43^{3} + 23\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 12\cdot 43 + 43^{2} + 31\cdot 43^{3} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 16\cdot 43 + 11\cdot 43^{2} + 7\cdot 43^{3} + 3\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 + 28\cdot 43 + 33\cdot 43^{2} + 11\cdot 43^{3} + 15\cdot 43^{4} +O(43^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ | |
| $1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ | |
| $1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ |