Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(505\)\(\medspace = 5 \cdot 101 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.643938125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | even |
| Determinant: | 1.505.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.4.2525.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 8x^{6} - 9x^{5} + 18x^{4} - 17x^{3} + 6x^{2} - 5x + 5 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 40\cdot 71 + 31\cdot 71^{2} + 11\cdot 71^{3} + 67\cdot 71^{4} + 50\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 47\cdot 71 + 21\cdot 71^{2} + 7\cdot 71^{3} + 63\cdot 71^{4} + 21\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 29\cdot 71 + 61\cdot 71^{2} + 3\cdot 71^{3} + 44\cdot 71^{4} + 19\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 54 + 5\cdot 71 + 26\cdot 71^{2} + 52\cdot 71^{3} + 30\cdot 71^{4} + 14\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 + 23\cdot 71 + 58\cdot 71^{2} + 2\cdot 71^{3} + 44\cdot 71^{4} + 61\cdot 71^{5} +O(71^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 68 + 66\cdot 71 + 22\cdot 71^{2} + 3\cdot 71^{3} + 57\cdot 71^{5} +O(71^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 69 + 48\cdot 71 + 62\cdot 71^{2} + 70\cdot 71^{3} + 51\cdot 71^{4} + 54\cdot 71^{5} +O(71^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 70 + 21\cdot 71 + 70\cdot 71^{2} + 60\cdot 71^{3} + 53\cdot 71^{4} + 3\cdot 71^{5} +O(71^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,5,4,8)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,7,8,3,4,2,5,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,7,4,6,8,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.