Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1040\)\(\medspace = 2^{4} \cdot 5 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1124864000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.260.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1040.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 12x^{6} - 18x^{5} + 19x^{4} - 10x^{3} + 6x^{2} + 6x + 1 \)
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The roots of $f$ are computed in $\Q_{ 137 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 62\cdot 137^{2} + 118\cdot 137^{3} + 65\cdot 137^{4} + 99\cdot 137^{5} +O(137^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 41 + 32\cdot 137 + 74\cdot 137^{2} + 53\cdot 137^{3} + 97\cdot 137^{4} + 130\cdot 137^{5} +O(137^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 56 + 134\cdot 137^{2} + 54\cdot 137^{3} + 136\cdot 137^{4} + 106\cdot 137^{5} +O(137^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 94 + 88\cdot 137 + 124\cdot 137^{2} + 90\cdot 137^{3} + 40\cdot 137^{4} + 41\cdot 137^{5} +O(137^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 108 + 84\cdot 137 + 29\cdot 137^{2} + 98\cdot 137^{3} + 7\cdot 137^{4} + 36\cdot 137^{5} +O(137^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 119 + 47\cdot 137 + 90\cdot 137^{2} + 9\cdot 137^{3} + 31\cdot 137^{4} + 26\cdot 137^{5} +O(137^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 130 + 3\cdot 137 + 20\cdot 137^{2} + 111\cdot 137^{3} + 98\cdot 137^{4} + 104\cdot 137^{5} +O(137^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 134 + 15\cdot 137 + 13\cdot 137^{2} + 11\cdot 137^{3} + 70\cdot 137^{4} + 2\cdot 137^{5} +O(137^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $-2$ |
| $4$ | $2$ | $(2,3)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,4,7)(2,6,8,3)$ | $0$ |
| $2$ | $8$ | $(1,8,5,3,4,2,7,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,7,8,4,6,5,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.