Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(2105\)\(\medspace = 5 \cdot 421 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.886205.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{421})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 8.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 10\cdot 11 + 6\cdot 11^{2} + 2\cdot 11^{3} + 3\cdot 11^{4} + 7\cdot 11^{5} + 11^{6} + 7\cdot 11^{7} +O(11^{8})\) |
$r_{ 2 }$ | $=$ | \( 5 + 10\cdot 11 + 9\cdot 11^{2} + 2\cdot 11^{3} + 6\cdot 11^{4} + 2\cdot 11^{5} + 5\cdot 11^{6} + 4\cdot 11^{7} +O(11^{8})\) |
$r_{ 3 }$ | $=$ | \( 6 + 11^{2} + 8\cdot 11^{3} + 4\cdot 11^{4} + 8\cdot 11^{5} + 5\cdot 11^{6} + 6\cdot 11^{7} +O(11^{8})\) |
$r_{ 4 }$ | $=$ | \( 8 + 4\cdot 11^{2} + 8\cdot 11^{3} + 7\cdot 11^{4} + 3\cdot 11^{5} + 9\cdot 11^{6} + 3\cdot 11^{7} +O(11^{8})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |