Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(544\)\(\medspace = 2^{5} \cdot 17 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.4.4352.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Determinant: | 1.136.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{17})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 6x^{2} - 4x + 2 \) . |
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 15 + 66\cdot 89 + 38\cdot 89^{2} + 73\cdot 89^{4} +O(89^{5})\)
$r_{ 2 }$ |
$=$ |
\( 29 + 2\cdot 89 + 72\cdot 89^{2} + 61\cdot 89^{3} + 8\cdot 89^{4} +O(89^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 49 + 88\cdot 89 + 10\cdot 89^{2} + 39\cdot 89^{3} + 79\cdot 89^{4} +O(89^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 85 + 20\cdot 89 + 56\cdot 89^{2} + 76\cdot 89^{3} + 16\cdot 89^{4} +O(89^{5})\)
| |
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.