Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(495\)\(\medspace = 3^{2} \cdot 5 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.741200625.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.55.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 4x^{6} - 14x^{5} + 29x^{4} - 28x^{3} + 36x^{2} - 55x + 55 \)
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The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 62\cdot 89 + 10\cdot 89^{2} + 13\cdot 89^{3} + 16\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 49\cdot 89 + 3\cdot 89^{2} + 3\cdot 89^{3} + 33\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 + 8\cdot 89 + 79\cdot 89^{2} + 79\cdot 89^{3} + 24\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 40 + 41\cdot 89 + 75\cdot 89^{2} + 32\cdot 89^{3} + 24\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 85\cdot 89 + 5\cdot 89^{2} + 20\cdot 89^{3} + 33\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 61 + 30\cdot 89 + 77\cdot 89^{2} + 15\cdot 89^{3} + 83\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 72 + 62\cdot 89 + 72\cdot 89^{2} + 52\cdot 89^{3} + 50\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 77 + 15\cdot 89 + 31\cdot 89^{2} + 49\cdot 89^{3} + 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,8,7,5)(2,3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.