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 8.0.134217728.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} + 10x^{4} - 8x^{2} + 2 \)
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The roots of $f$ are computed in $\Q_{ 137 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 28\cdot 137 + 18\cdot 137^{2} + 10\cdot 137^{3} + 59\cdot 137^{4} + 86\cdot 137^{5} +O(137^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 22 + 59\cdot 137 + 64\cdot 137^{2} + 88\cdot 137^{3} + 18\cdot 137^{4} + 80\cdot 137^{5} +O(137^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 27 + 44\cdot 137 + 96\cdot 137^{2} + 4\cdot 137^{3} + 133\cdot 137^{4} + 134\cdot 137^{5} +O(137^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 35 + 69\cdot 137 + 56\cdot 137^{2} + 42\cdot 137^{3} + 64\cdot 137^{4} + 73\cdot 137^{5} +O(137^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 102 + 67\cdot 137 + 80\cdot 137^{2} + 94\cdot 137^{3} + 72\cdot 137^{4} + 63\cdot 137^{5} +O(137^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 110 + 92\cdot 137 + 40\cdot 137^{2} + 132\cdot 137^{3} + 3\cdot 137^{4} + 2\cdot 137^{5} +O(137^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 115 + 77\cdot 137 + 72\cdot 137^{2} + 48\cdot 137^{3} + 118\cdot 137^{4} + 56\cdot 137^{5} +O(137^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 124 + 108\cdot 137 + 118\cdot 137^{2} + 126\cdot 137^{3} + 77\cdot 137^{4} + 50\cdot 137^{5} +O(137^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,7,6,4,8,2,3,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.