Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(85\)\(\medspace = 5 \cdot 17 \) |
| Artin field: | Galois closure of 4.4.122825.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{85}(64,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 23x^{2} + x + 86 \)
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The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 41\cdot 43 + 5\cdot 43^{2} + 43^{3} + 5\cdot 43^{4} +O(43^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 4 + 19\cdot 43 + 20\cdot 43^{2} + 31\cdot 43^{4} +O(43^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 6 + 42\cdot 43 + 34\cdot 43^{3} + 29\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 + 26\cdot 43 + 15\cdot 43^{2} + 7\cdot 43^{3} + 20\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,2,4,3)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,3,4,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.