Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(539\)\(\medspace = 7^{2} \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.1722499009.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-7}, \sqrt{-11})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + x^{6} - 11x^{4} + 4x^{2} + 16 \)
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The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 24\cdot 37 + 8\cdot 37^{2} + 28\cdot 37^{3} + 10\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 13\cdot 37 + 22\cdot 37^{2} + 19\cdot 37^{3} + 15\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 6\cdot 37 + 14\cdot 37^{2} + 20\cdot 37^{3} + 7\cdot 37^{4} +O(37^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 18 + 21\cdot 37 + 37^{2} + 25\cdot 37^{3} + 2\cdot 37^{4} +O(37^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 19 + 15\cdot 37 + 35\cdot 37^{2} + 11\cdot 37^{3} + 34\cdot 37^{4} +O(37^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 22 + 30\cdot 37 + 22\cdot 37^{2} + 16\cdot 37^{3} + 29\cdot 37^{4} +O(37^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 27 + 23\cdot 37 + 14\cdot 37^{2} + 17\cdot 37^{3} + 21\cdot 37^{4} +O(37^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 33 + 12\cdot 37 + 28\cdot 37^{2} + 8\cdot 37^{3} + 26\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,4,7)(2,8,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.