Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.95004009.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-19})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + x^{6} + 6x^{5} + 6x^{4} + 6x^{3} + x^{2} - 3x + 1 \)
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The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 41\cdot 43 + 17\cdot 43^{2} + 40\cdot 43^{3} + 2\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 37\cdot 43 + 23\cdot 43^{2} + 7\cdot 43^{3} + 26\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 19\cdot 43^{2} + 25\cdot 43^{3} + 7\cdot 43^{4} +O(43^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 30 + 18\cdot 43 + 38\cdot 43^{2} + 6\cdot 43^{3} + 3\cdot 43^{4} +O(43^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 32 + 14\cdot 43 + 4\cdot 43^{2} + 31\cdot 43^{3} + 18\cdot 43^{4} +O(43^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 33 + 21\cdot 43 + 12\cdot 43^{2} + 34\cdot 43^{3} + 43^{4} +O(43^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 35 + 15\cdot 43 + 31\cdot 43^{2} + 15\cdot 43^{3} + 21\cdot 43^{4} +O(43^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 39 + 21\cdot 43 + 24\cdot 43^{2} + 10\cdot 43^{3} + 4\cdot 43^{4} +O(43^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,6,8,7)(2,3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.