Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.157351936.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 2x^{6} + 10x^{4} + 2x^{2} + 1 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 18\cdot 71 + 3\cdot 71^{2} + 9\cdot 71^{3} + 41\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 35\cdot 71 + 57\cdot 71^{2} + 6\cdot 71^{3} + 67\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 48\cdot 71 + 26\cdot 71^{2} + 61\cdot 71^{3} + 24\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 46\cdot 71 + 14\cdot 71^{2} + 15\cdot 71^{3} + 21\cdot 71^{4} +O(71^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 46 + 24\cdot 71 + 56\cdot 71^{2} + 55\cdot 71^{3} + 49\cdot 71^{4} +O(71^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 54 + 22\cdot 71 + 44\cdot 71^{2} + 9\cdot 71^{3} + 46\cdot 71^{4} +O(71^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 61 + 35\cdot 71 + 13\cdot 71^{2} + 64\cdot 71^{3} + 3\cdot 71^{4} +O(71^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 64 + 52\cdot 71 + 67\cdot 71^{2} + 61\cdot 71^{3} + 29\cdot 71^{4} +O(71^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,5,2,6)(3,8,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.