Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1776\)\(\medspace = 2^{4} \cdot 3 \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1050340608.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.111.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.333.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 8x^{6} + 18x^{4} + 17x^{2} + 37 \)
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The roots of $f$ are computed in $\Q_{ 151 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 30 + 70\cdot 151 + 57\cdot 151^{2} + 13\cdot 151^{3} + 63\cdot 151^{4} + 37\cdot 151^{5} +O(151^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 + 47\cdot 151 + 85\cdot 151^{2} + 66\cdot 151^{3} + 135\cdot 151^{4} +O(151^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 38 + 92\cdot 151 + 25\cdot 151^{2} + 140\cdot 151^{3} + 53\cdot 151^{4} + 42\cdot 151^{5} +O(151^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 60 + 9\cdot 151 + 98\cdot 151^{2} + 135\cdot 151^{3} + 80\cdot 151^{4} + 4\cdot 151^{5} +O(151^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 91 + 141\cdot 151 + 52\cdot 151^{2} + 15\cdot 151^{3} + 70\cdot 151^{4} + 146\cdot 151^{5} +O(151^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 113 + 58\cdot 151 + 125\cdot 151^{2} + 10\cdot 151^{3} + 97\cdot 151^{4} + 108\cdot 151^{5} +O(151^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 115 + 103\cdot 151 + 65\cdot 151^{2} + 84\cdot 151^{3} + 15\cdot 151^{4} + 150\cdot 151^{5} +O(151^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 121 + 80\cdot 151 + 93\cdot 151^{2} + 137\cdot 151^{3} + 87\cdot 151^{4} + 113\cdot 151^{5} +O(151^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,8)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.