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 8.0.339738624.10 |
| 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 $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 4\cdot 67 + 27\cdot 67^{2} + 18\cdot 67^{3} + 3\cdot 67^{4} +O(67^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 9 + 64\cdot 67 + 67^{2} + 47\cdot 67^{3} + 50\cdot 67^{4} +O(67^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 12 + 31\cdot 67 + 26\cdot 67^{2} + 12\cdot 67^{3} + 24\cdot 67^{4} +O(67^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 13 + 19\cdot 67 + 3\cdot 67^{2} + 25\cdot 67^{3} + 4\cdot 67^{4} +O(67^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 54 + 47\cdot 67 + 63\cdot 67^{2} + 41\cdot 67^{3} + 62\cdot 67^{4} +O(67^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 55 + 35\cdot 67 + 40\cdot 67^{2} + 54\cdot 67^{3} + 42\cdot 67^{4} +O(67^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 58 + 2\cdot 67 + 65\cdot 67^{2} + 19\cdot 67^{3} + 16\cdot 67^{4} +O(67^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 65 + 62\cdot 67 + 39\cdot 67^{2} + 48\cdot 67^{3} + 63\cdot 67^{4} +O(67^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.