Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(95\)\(\medspace = 5 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.81450625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.95.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-19})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 8x^{6} + 7x^{5} + 19x^{4} + 7x^{3} + 8x^{2} - x + 1 \)
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The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 + 51\cdot 131 + 125\cdot 131^{2} + 35\cdot 131^{3} + 8\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 91\cdot 131 + 30\cdot 131^{2} + 127\cdot 131^{3} + 16\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 55 + 30\cdot 131 + 74\cdot 131^{2} + 52\cdot 131^{3} + 98\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 58 + 5\cdot 131 + 117\cdot 131^{2} + 118\cdot 131^{3} + 53\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 61 + 53\cdot 131 + 21\cdot 131^{2} + 19\cdot 131^{3} + 118\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 81 + 60\cdot 131 + 79\cdot 131^{2} + 48\cdot 131^{3} + 79\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 86 + 70\cdot 131 + 115\cdot 131^{2} + 49\cdot 131^{3} + 36\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 126 + 29\cdot 131 + 91\cdot 131^{2} + 71\cdot 131^{3} + 112\cdot 131^{4} +O(131^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,6,5)(3,4,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.