Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(41\) |
| Artin field: | Galois closure of 4.4.68921.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{41}(9,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 15x^{2} - 18x - 4 \)
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The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 15\cdot 23 + 9\cdot 23^{2} + 5\cdot 23^{3} + 13\cdot 23^{4} +O(23^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 11 + 10\cdot 23 + 16\cdot 23^{2} + 21\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 23 + 8\cdot 23^{2} + 6\cdot 23^{3} + 14\cdot 23^{4} +O(23^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 20 + 18\cdot 23 + 11\cdot 23^{2} + 12\cdot 23^{3} + 7\cdot 23^{4} +O(23^{5})\)
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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.