Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(128\)\(\medspace = 2^{7} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.512.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.8.2t1.b.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\zeta_{8})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 2x^{2} + 2 \) . |
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 16 + 27\cdot 41 + 34\cdot 41^{2} + 6\cdot 41^{3} + 17\cdot 41^{4} +O(41^{5})\)
$r_{ 2 }$ |
$=$ |
\( 19 + 6\cdot 41 + 39\cdot 41^{2} + 12\cdot 41^{3} + 38\cdot 41^{4} +O(41^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 22 + 34\cdot 41 + 41^{2} + 28\cdot 41^{3} + 2\cdot 41^{4} +O(41^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 25 + 13\cdot 41 + 6\cdot 41^{2} + 34\cdot 41^{3} + 23\cdot 41^{4} +O(41^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.