Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(6371\)\(\medspace = 23 \cdot 277 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.1764767.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.6371.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-23}, \sqrt{277})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} - 31x^{2} + 32x - 21 \)
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The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 19\cdot 29 + 13\cdot 29^{2} + 24\cdot 29^{3} + 27\cdot 29^{4} + 16\cdot 29^{5} +O(29^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 5 + 11\cdot 29 + 16\cdot 29^{2} + 7\cdot 29^{3} + 25\cdot 29^{4} + 4\cdot 29^{5} +O(29^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 25 + 17\cdot 29 + 12\cdot 29^{2} + 21\cdot 29^{3} + 3\cdot 29^{4} + 24\cdot 29^{5} +O(29^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 26 + 9\cdot 29 + 15\cdot 29^{2} + 4\cdot 29^{3} + 29^{4} + 12\cdot 29^{5} +O(29^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.