Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(248\)\(\medspace = 2^{3} \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.472842752.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.248.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.1984.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 3x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 3x^{2} + 3x + 1 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 36\cdot 71 + 43\cdot 71^{2} + 55\cdot 71^{3} + 68\cdot 71^{4} + 28\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 58\cdot 71 + 59\cdot 71^{2} + 14\cdot 71^{3} + 26\cdot 71^{4} + 9\cdot 71^{5} +O(71^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 40 + 34\cdot 71 + 71^{2} + 9\cdot 71^{3} + 30\cdot 71^{4} + 49\cdot 71^{5} +O(71^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 55 + 43\cdot 71 + 46\cdot 71^{2} + 35\cdot 71^{3} + 11\cdot 71^{4} + 40\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 + 45\cdot 71 + 22\cdot 71^{2} + 61\cdot 71^{3} + 61\cdot 71^{4} + 11\cdot 71^{5} +O(71^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 57 + 34\cdot 71 + 26\cdot 71^{2} + 17\cdot 71^{3} + 30\cdot 71^{4} + 66\cdot 71^{5} +O(71^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 59 + 12\cdot 71 + 68\cdot 71^{2} + 34\cdot 71^{3} + 65\cdot 71^{4} + 2\cdot 71^{5} +O(71^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 66 + 17\cdot 71 + 15\cdot 71^{2} + 55\cdot 71^{3} + 60\cdot 71^{4} + 3\cdot 71^{5} +O(71^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,4)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(3,4)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,6,7,8)(2,3,5,4)$ | $0$ |
| $2$ | $8$ | $(1,5,8,3,7,2,6,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,6,5,7,4,8,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.