Basic invariants
| Dimension: | $1$ |
| Group: | $C_4$ |
| Conductor: | \(65\)\(\medspace = 5 \cdot 13 \) |
| Artin field: | Galois closure of 4.0.21125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{65}(12,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 16x^{2} - 16x + 61 \)
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The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 43\cdot 59 + 3\cdot 59^{2} + 23\cdot 59^{3} + 9\cdot 59^{4} +O(59^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 14 + 10\cdot 59 + 22\cdot 59^{2} + 22\cdot 59^{3} + 27\cdot 59^{4} +O(59^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 43 + 10\cdot 59 + 27\cdot 59^{2} + 59^{3} + 17\cdot 59^{4} +O(59^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 50 + 53\cdot 59 + 5\cdot 59^{2} + 12\cdot 59^{3} + 5\cdot 59^{4} +O(59^{5})\)
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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.