Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(505\)\(\medspace = 5 \cdot 101 \) |
| Artin field: | Galois closure of 4.0.1275125.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{505}(403,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 126x^{2} - 126x + 3251 \)
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The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 26\cdot 29 + 22\cdot 29^{2} + 2\cdot 29^{3} + 25\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 22\cdot 29 + 3\cdot 29^{2} + 14\cdot 29^{3} + 11\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 14\cdot 29 + 20\cdot 29^{2} + 15\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 23\cdot 29 + 10\cdot 29^{2} + 25\cdot 29^{3} + 18\cdot 29^{4} +O(29^{5})\)
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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.