Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(85\)\(\medspace = 5 \cdot 17 \) |
| Artin field: | Galois closure of 4.0.36125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{85}(67,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 21x^{2} - 21x + 101 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 75\cdot 79 + 67\cdot 79^{2} + 47\cdot 79^{3} + 66\cdot 79^{4} +O(79^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 19 + 48\cdot 79 + 66\cdot 79^{2} + 63\cdot 79^{3} + 20\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 63 + 7\cdot 79 + 48\cdot 79^{2} + 29\cdot 79^{3} + 55\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 66 + 26\cdot 79 + 54\cdot 79^{2} + 16\cdot 79^{3} + 15\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.