Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(1205\)\(\medspace = 5 \cdot 241 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.290405.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{241})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 8 + 8\cdot 41 + 11\cdot 41^{2} + 2\cdot 41^{3} + 8\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 12 + 32\cdot 41 + 7\cdot 41^{2} + 36\cdot 41^{3} + 9\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 30 + 8\cdot 41 + 33\cdot 41^{2} + 4\cdot 41^{3} + 31\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 34 + 32\cdot 41 + 29\cdot 41^{2} + 38\cdot 41^{3} + 32\cdot 41^{4} +O(41^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |