Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(689871\)\(\medspace = 3 \cdot 7^{2} \cdot 13 \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.8968323.5 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.39.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 32x^{2} - 232x - 3233 \)
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The roots of $f$ are computed in $\Q_{ 43 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 + 35\cdot 43 + 33\cdot 43^{2} + 40\cdot 43^{3} + 13\cdot 43^{4} + 42\cdot 43^{5} +O(43^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 28 + 5\cdot 43 + 7\cdot 43^{2} + 26\cdot 43^{3} + 10\cdot 43^{4} + 7\cdot 43^{5} +O(43^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 33 + 14\cdot 43 + 40\cdot 43^{2} + 29\cdot 43^{3} + 20\cdot 43^{4} + 43^{5} +O(43^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 42 + 29\cdot 43 + 4\cdot 43^{2} + 32\cdot 43^{3} + 40\cdot 43^{4} + 34\cdot 43^{5} +O(43^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.