Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(3104\)\(\medspace = 2^{5} \cdot 97 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.5801854959616.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.776.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{97})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{6} - 7x^{4} + 784x^{2} + 16 \)
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The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 63\cdot 89 + 41\cdot 89^{2} + 13\cdot 89^{3} + 5\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 58\cdot 89 + 10\cdot 89^{2} + 6\cdot 89^{3} + 2\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 38\cdot 89 + 41\cdot 89^{2} + 40\cdot 89^{3} + 9\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 32 + 33\cdot 89 + 10\cdot 89^{2} + 33\cdot 89^{3} + 6\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 + 55\cdot 89 + 78\cdot 89^{2} + 55\cdot 89^{3} + 82\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 66 + 50\cdot 89 + 47\cdot 89^{2} + 48\cdot 89^{3} + 79\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 68 + 30\cdot 89 + 78\cdot 89^{2} + 82\cdot 89^{3} + 86\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 77 + 25\cdot 89 + 47\cdot 89^{2} + 75\cdot 89^{3} + 83\cdot 89^{4} +O(89^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ | |
| $2$ | $4$ | $(1,4,7,6)(2,3,8,5)$ | $0$ |