Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(512\)\(\medspace = 2^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.268435456.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.512.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{4} - 8x^{2} - 1 \)
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The roots of $f$ are computed in $\Q_{ 137 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 + 81\cdot 137 + 51\cdot 137^{2} + 52\cdot 137^{3} + 72\cdot 137^{4} + 48\cdot 137^{5} +O(137^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 39 + 33\cdot 137 + 49\cdot 137^{2} + 69\cdot 137^{4} + 78\cdot 137^{5} +O(137^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 44 + 137 + 95\cdot 137^{2} + 113\cdot 137^{3} + 103\cdot 137^{4} + 83\cdot 137^{5} +O(137^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 46 + 63\cdot 137 + 29\cdot 137^{2} + 11\cdot 137^{3} + 34\cdot 137^{4} + 62\cdot 137^{5} +O(137^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 91 + 73\cdot 137 + 107\cdot 137^{2} + 125\cdot 137^{3} + 102\cdot 137^{4} + 74\cdot 137^{5} +O(137^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 93 + 135\cdot 137 + 41\cdot 137^{2} + 23\cdot 137^{3} + 33\cdot 137^{4} + 53\cdot 137^{5} +O(137^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 98 + 103\cdot 137 + 87\cdot 137^{2} + 136\cdot 137^{3} + 67\cdot 137^{4} + 58\cdot 137^{5} +O(137^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 110 + 55\cdot 137 + 85\cdot 137^{2} + 84\cdot 137^{3} + 64\cdot 137^{4} + 88\cdot 137^{5} +O(137^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.