Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(612\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.1731891456.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.68.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 3x^{6} + 11x^{4} + 45x^{2} + 4 \)
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The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 + 8\cdot 89 + 32\cdot 89^{2} + 36\cdot 89^{3} + 10\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 13\cdot 89 + 22\cdot 89^{2} + 46\cdot 89^{3} + 21\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 + 2\cdot 89 + 80\cdot 89^{2} + 4\cdot 89^{3} + 52\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 + 26\cdot 89 + 71\cdot 89^{2} + 39\cdot 89^{3} + 54\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 48 + 62\cdot 89 + 17\cdot 89^{2} + 49\cdot 89^{3} + 34\cdot 89^{4} +O(89^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 61 + 86\cdot 89 + 8\cdot 89^{2} + 84\cdot 89^{3} + 36\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 64 + 75\cdot 89 + 66\cdot 89^{2} + 42\cdot 89^{3} + 67\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 72 + 80\cdot 89 + 56\cdot 89^{2} + 52\cdot 89^{3} + 78\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,4,7)(2,8,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.