Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.1197.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.399.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{133})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} - 4x^{2} + 5x + 7 \)
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The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 10\cdot 13 + 4\cdot 13^{2} + 10\cdot 13^{3} + 3\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 2\cdot 13 + 2\cdot 13^{2} + 13^{3} + 5\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 10\cdot 13 + 10\cdot 13^{2} + 11\cdot 13^{3} + 7\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 2\cdot 13 + 8\cdot 13^{2} + 2\cdot 13^{3} + 9\cdot 13^{4} +O(13^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.