Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(1680\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.33600.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.420.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{21})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 8x^{2} - 2x + 1 \)
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The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 14\cdot 41 + 30\cdot 41^{2} + 29\cdot 41^{3} + 3\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 14\cdot 41 + 27\cdot 41^{2} + 21\cdot 41^{3} + 38\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 9\cdot 41 + 14\cdot 41^{2} + 7\cdot 41^{3} + 29\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 2\cdot 41 + 10\cdot 41^{2} + 23\cdot 41^{3} + 10\cdot 41^{4} +O(41^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,3)(2,4)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,3)$ | $0$ | |
| $2$ | $4$ | $(1,4,3,2)$ | $0$ |