Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(260\)\(\medspace = 2^{2} \cdot 5 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.4569760000.4 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.260.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{65})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} + 29x^{4} - 62x^{3} + 18x^{2} - 84x + 196 \)
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The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 84\cdot 101 + 74\cdot 101^{2} + 58\cdot 101^{3} + 49\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 23\cdot 101 + 85\cdot 101^{2} + 45\cdot 101^{3} + 86\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 + 65\cdot 101 + 91\cdot 101^{2} + 44\cdot 101^{3} + 84\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 71 + 40\cdot 101 + 38\cdot 101^{2} + 21\cdot 101^{3} + 27\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 82 + 11\cdot 101 + 98\cdot 101^{2} + 76\cdot 101^{3} + 40\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 95 + 49\cdot 101 + 92\cdot 101^{2} + 44\cdot 101^{3} + 30\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 98 + 45\cdot 101 + 80\cdot 101^{2} + 90\cdot 101^{3} + 59\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 100 + 82\cdot 101 + 44\cdot 101^{2} + 20\cdot 101^{3} + 25\cdot 101^{4} +O(101^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,5,4)(2,7,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.