Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.9150625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.55.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 9x^{6} - 13x^{5} + 18x^{4} - 11x^{3} + 11x^{2} - 4x + 1 \)
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The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 56\cdot 59 + 31\cdot 59^{2} + 6\cdot 59^{3} + 27\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 35\cdot 59 + 54\cdot 59^{2} + 48\cdot 59^{3} + 58\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 54\cdot 59 + 47\cdot 59^{2} + 46\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 35\cdot 59 + 43\cdot 59^{2} + 34\cdot 59^{3} + 38\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 + 24\cdot 59 + 44\cdot 59^{2} + 30\cdot 59^{3} + 19\cdot 59^{4} +O(59^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 28 + 14\cdot 59 + 18\cdot 59^{2} + 46\cdot 59^{3} + 37\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 37 + 19\cdot 59 + 41\cdot 59^{2} + 49\cdot 59^{3} + 17\cdot 59^{4} +O(59^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 44 + 55\cdot 59 + 12\cdot 59^{2} + 18\cdot 59^{3} + 49\cdot 59^{4} +O(59^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,7,2,6)(3,5,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.