Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.448084224.6 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.7.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 3x^{6} + 12x^{4} + 9x^{2} + 9 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 26\cdot 43 + 26\cdot 43^{2} + 23\cdot 43^{3} + 24\cdot 43^{4} +O(43^{5})\)
$r_{ 2 }$ |
$=$ |
\( 12 + 7\cdot 43 + 23\cdot 43^{2} + 27\cdot 43^{3} + 29\cdot 43^{4} +O(43^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 15 + 8\cdot 43 + 38\cdot 43^{3} + 7\cdot 43^{4} +O(43^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 19 + 37\cdot 43 + 30\cdot 43^{2} + 21\cdot 43^{3} + 15\cdot 43^{4} +O(43^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 24 + 5\cdot 43 + 12\cdot 43^{2} + 21\cdot 43^{3} + 27\cdot 43^{4} +O(43^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 28 + 34\cdot 43 + 42\cdot 43^{2} + 4\cdot 43^{3} + 35\cdot 43^{4} +O(43^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 31 + 35\cdot 43 + 19\cdot 43^{2} + 15\cdot 43^{3} + 13\cdot 43^{4} +O(43^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 41 + 16\cdot 43 + 16\cdot 43^{2} + 19\cdot 43^{3} + 18\cdot 43^{4} +O(43^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
$2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
$2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.