Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.21381376.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.68.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 5x^{6} + 4x^{4} + 5x^{2} + 1 \)
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The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 48\cdot 53 + 9\cdot 53^{2} + 48\cdot 53^{3} + 49\cdot 53^{4} +O(53^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 11 + 30\cdot 53 + 50\cdot 53^{2} + 12\cdot 53^{3} + 53^{4} +O(53^{5})\)
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| $r_{ 3 }$ | $=$ |
\( 20 + 32\cdot 53 + 41\cdot 53^{2} + 37\cdot 53^{3} + 45\cdot 53^{4} +O(53^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 24 + 16\cdot 53 + 24\cdot 53^{2} + 25\cdot 53^{3} + 50\cdot 53^{4} +O(53^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 29 + 36\cdot 53 + 28\cdot 53^{2} + 27\cdot 53^{3} + 2\cdot 53^{4} +O(53^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 33 + 20\cdot 53 + 11\cdot 53^{2} + 15\cdot 53^{3} + 7\cdot 53^{4} +O(53^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 42 + 22\cdot 53 + 2\cdot 53^{2} + 40\cdot 53^{3} + 51\cdot 53^{4} +O(53^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 45 + 4\cdot 53 + 43\cdot 53^{2} + 4\cdot 53^{3} + 3\cdot 53^{4} +O(53^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,5)(2,3,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.