Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(3479\)\(\medspace = 7^{2} \cdot 71 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.2.1729063.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-7}, \sqrt{-71})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 5 + 22\cdot 43 + 8\cdot 43^{2} + 2\cdot 43^{3} + 38\cdot 43^{4} +O(43^{5})\)
$r_{ 2 }$ |
$=$ |
\( 21 + 16\cdot 43 + 24\cdot 43^{2} + 22\cdot 43^{3} + 12\cdot 43^{4} +O(43^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 22 + 26\cdot 43 + 18\cdot 43^{2} + 20\cdot 43^{3} + 30\cdot 43^{4} +O(43^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 38 + 20\cdot 43 + 34\cdot 43^{2} + 40\cdot 43^{3} + 4\cdot 43^{4} +O(43^{5})\)
| |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |