Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.1358954496.9 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\zeta_{8})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} + 72x^{4} - 108x^{2} + 81 \)
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The roots of $f$ are computed in $\Q_{ 17 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 4\cdot 17 + 2\cdot 17^{3} + 10\cdot 17^{4} + 16\cdot 17^{5} +O(17^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 2 + 6\cdot 17^{2} + 13\cdot 17^{3} + 11\cdot 17^{4} + 13\cdot 17^{5} +O(17^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 3 + 5\cdot 17 + 13\cdot 17^{2} + 8\cdot 17^{3} + 9\cdot 17^{4} + 5\cdot 17^{5} +O(17^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 7 + 8\cdot 17 + 4\cdot 17^{2} + 14\cdot 17^{3} + 3\cdot 17^{4} + 6\cdot 17^{5} +O(17^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 10 + 8\cdot 17 + 12\cdot 17^{2} + 2\cdot 17^{3} + 13\cdot 17^{4} + 10\cdot 17^{5} +O(17^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 14 + 11\cdot 17 + 3\cdot 17^{2} + 8\cdot 17^{3} + 7\cdot 17^{4} + 11\cdot 17^{5} +O(17^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 15 + 16\cdot 17 + 10\cdot 17^{2} + 3\cdot 17^{3} + 5\cdot 17^{4} + 3\cdot 17^{5} +O(17^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 16 + 12\cdot 17 + 16\cdot 17^{2} + 14\cdot 17^{3} + 6\cdot 17^{4} +O(17^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.