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 8.2.122023936.2 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 3x^{6} + 6x^{5} + 6x^{4} - 6x^{3} - 5x^{2} - 2x + 1 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 60\cdot 71 + 4\cdot 71^{2} + 25\cdot 71^{3} + 11\cdot 71^{4} + 30\cdot 71^{5} +O(71^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 11 + 62\cdot 71 + 51\cdot 71^{2} + 50\cdot 71^{3} + 61\cdot 71^{4} + 71^{5} +O(71^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 21 + 7\cdot 71 + 33\cdot 71^{2} + 25\cdot 71^{3} + 19\cdot 71^{4} + 55\cdot 71^{5} +O(71^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 31 + 63\cdot 71 + 68\cdot 71^{2} + 17\cdot 71^{3} + 29\cdot 71^{4} + 35\cdot 71^{5} +O(71^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 37 + 18\cdot 71 + 54\cdot 71^{2} + 6\cdot 71^{3} + 51\cdot 71^{4} + 17\cdot 71^{5} +O(71^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 44 + 23\cdot 71 + 71^{2} + 27\cdot 71^{3} + 46\cdot 71^{4} + 18\cdot 71^{5} +O(71^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 65 + 70\cdot 71 + 13\cdot 71^{2} + 21\cdot 71^{3} + 50\cdot 71^{4} + 58\cdot 71^{5} +O(71^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 67 + 48\cdot 71 + 55\cdot 71^{2} + 38\cdot 71^{3} + 14\cdot 71^{4} + 66\cdot 71^{5} +O(71^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,4)(5,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,4)(3,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,7,4)(2,3,6,8)$ | $0$ |
| $2$ | $8$ | $(1,2,4,8,7,6,5,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,8,5,2,7,3,4,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.