Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.20123648.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.68.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.272.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 2x^{6} + 2x^{5} + 6x^{4} - 6x^{3} + 6x^{2} - 2x + 1 \)
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The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 45\cdot 149 + 19\cdot 149^{2} + 12\cdot 149^{3} + 127\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 66\cdot 149 + 22\cdot 149^{2} + 90\cdot 149^{3} + 45\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 24\cdot 149 + 58\cdot 149^{2} + 63\cdot 149^{3} + 41\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 + 7\cdot 149 + 47\cdot 149^{2} + 43\cdot 149^{3} + 127\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 + 137\cdot 149 + 80\cdot 149^{2} + 15\cdot 149^{3} + 100\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 69 + 4\cdot 149 + 142\cdot 149^{2} + 43\cdot 149^{3} + 140\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 140 + 122\cdot 149 + 21\cdot 149^{2} + 134\cdot 149^{3} + 92\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 142 + 39\cdot 149 + 55\cdot 149^{2} + 44\cdot 149^{3} + 70\cdot 149^{4} +O(149^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,6,7)(2,4,8,3)$ | $0$ |
| $2$ | $8$ | $(1,3,5,2,6,4,7,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,7,3,6,8,5,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.