Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.64000000.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} - 10x^{5} + 11x^{4} - 10x^{3} + 2x^{2} + 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 17\cdot 101 + 53\cdot 101^{2} + 38\cdot 101^{3} + 34\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 89\cdot 101 + 76\cdot 101^{2} + 27\cdot 101^{3} + 77\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 + 84\cdot 101 + 97\cdot 101^{2} + 29\cdot 101^{3} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 + 89\cdot 101 + 25\cdot 101^{2} + 96\cdot 101^{3} + 10\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 61 + 11\cdot 101 + 75\cdot 101^{2} + 4\cdot 101^{3} + 90\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 71 + 16\cdot 101 + 3\cdot 101^{2} + 71\cdot 101^{3} + 100\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 95 + 11\cdot 101 + 24\cdot 101^{2} + 73\cdot 101^{3} + 23\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 98 + 83\cdot 101 + 47\cdot 101^{2} + 62\cdot 101^{3} + 66\cdot 101^{4} +O(101^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.