Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.0.513.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-19})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 2\cdot 43 + 23\cdot 43^{2} + 25\cdot 43^{3} + 35\cdot 43^{4} +O(43^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 19 + 7\cdot 43 + 43^{2} + 38\cdot 43^{3} + 13\cdot 43^{4} +O(43^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 20 + 12\cdot 43 + 7\cdot 43^{2} + 39\cdot 43^{3} + 2\cdot 43^{4} +O(43^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 30 + 20\cdot 43 + 11\cdot 43^{2} + 26\cdot 43^{3} + 33\cdot 43^{4} +O(43^{5})\)
| |
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |