Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(975\)\(\medspace = 3 \cdot 5^{2} \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.2925.1 |
| 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
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 8x^{2} + 6x + 21 \)
|
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 + 34\cdot 127 + 20\cdot 127^{2} + 61\cdot 127^{3} + 3\cdot 127^{4} +O(127^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 49 + 33\cdot 127 + 27\cdot 127^{2} + 74\cdot 127^{3} + 98\cdot 127^{4} +O(127^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 73 + 53\cdot 127 + 116\cdot 127^{2} + 55\cdot 127^{3} + 25\cdot 127^{4} +O(127^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 98 + 5\cdot 127 + 90\cdot 127^{2} + 62\cdot 127^{3} + 126\cdot 127^{4} +O(127^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,3)$ | $0$ |
| $2$ | $4$ | $(1,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.