Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.320.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(i, \sqrt{5})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 3 + 22\cdot 29 + 9\cdot 29^{2} + 12\cdot 29^{3} + 15\cdot 29^{4} +O(29^{5})\)
|
$r_{ 2 }$ | $=$ |
\( 5 + 8\cdot 29 + 12\cdot 29^{2} + 7\cdot 29^{3} + 9\cdot 29^{4} +O(29^{5})\)
|
$r_{ 3 }$ | $=$ |
\( 8 + 22\cdot 29 + 28\cdot 29^{2} + 22\cdot 29^{3} + 8\cdot 29^{4} +O(29^{5})\)
|
$r_{ 4 }$ | $=$ |
\( 15 + 5\cdot 29 + 7\cdot 29^{2} + 15\cdot 29^{3} + 24\cdot 29^{4} +O(29^{5})\)
|
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |