Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(205\)\(\medspace = 5 \cdot 41 \) |
| Artin field: | Galois closure of 4.0.210125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{205}(122,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 51x^{2} - 51x + 551 \)
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The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11\cdot 19 + 18\cdot 19^{2} + 2\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 1 + 2\cdot 19 + 18\cdot 19^{2} + 8\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 5 + 10\cdot 19 + 10\cdot 19^{2} + 11\cdot 19^{3} + 14\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 + 14\cdot 19 + 9\cdot 19^{2} + 14\cdot 19^{3} + 18\cdot 19^{4} +O(19^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $1$ | $4$ | $(1,4,3,2)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,2,3,4)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.