Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(136\)\(\medspace = 2^{3} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.342102016.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.136.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 3x^{6} + 20x^{5} + 3x^{4} - 2x^{3} + 9x^{2} - 8x + 8 \)
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The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 31\cdot 89 + 4\cdot 89^{2} + 47\cdot 89^{3} + 68\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 35\cdot 89 + 20\cdot 89^{2} + 74\cdot 89^{3} + 54\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 7\cdot 89 + 6\cdot 89^{2} + 64\cdot 89^{3} + 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 + 68\cdot 89 + 15\cdot 89^{2} + 29\cdot 89^{3} + 87\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 52 + 18\cdot 89 + 16\cdot 89^{2} + 82\cdot 89^{3} + 81\cdot 89^{4} +O(89^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 54 + 20\cdot 89 + 30\cdot 89^{3} + 20\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 69 + 57\cdot 89 + 68\cdot 89^{2} + 71\cdot 89^{3} + 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 76 + 27\cdot 89 + 46\cdot 89^{2} + 46\cdot 89^{3} + 39\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,5,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.