Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.339738624.9 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{-3})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} + 10x^{4} - 24x^{2} + 36 \)
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The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 4\cdot 19 + 8\cdot 19^{2} + 7\cdot 19^{4} +O(19^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 3 + 19 + 5\cdot 19^{2} + 9\cdot 19^{3} + 16\cdot 19^{4} +O(19^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 5 + 16\cdot 19 + 2\cdot 19^{2} + 6\cdot 19^{3} + 5\cdot 19^{4} +O(19^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 8 + 9\cdot 19 + 3\cdot 19^{3} + 7\cdot 19^{4} +O(19^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 11 + 9\cdot 19 + 18\cdot 19^{2} + 15\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 14 + 2\cdot 19 + 16\cdot 19^{2} + 12\cdot 19^{3} + 13\cdot 19^{4} +O(19^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 16 + 17\cdot 19 + 13\cdot 19^{2} + 9\cdot 19^{3} + 2\cdot 19^{4} +O(19^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 18 + 14\cdot 19 + 10\cdot 19^{2} + 18\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.