Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(305\)\(\medspace = 5 \cdot 61 \) |
| Artin field: | Galois closure of 4.0.465125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{305}(243,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 76x^{2} - 76x + 1201 \)
|
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 10\cdot 29 + 27\cdot 29^{2} + 27\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 26\cdot 29 + 12\cdot 29^{2} + 26\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 23\cdot 29 + 20\cdot 29^{2} + 29^{3} + 3\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 + 26\cdot 29 + 25\cdot 29^{2} + 29^{3} + 7\cdot 29^{4} +O(29^{5})\)
|
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.