Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(328\)\(\medspace = 2^{3} \cdot 41 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 4.0.2624.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{41})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 13 + 82\cdot 83 + 9\cdot 83^{2} + 30\cdot 83^{3} + 74\cdot 83^{4} +O(83^{5})\) |
$r_{ 2 }$ | $=$ | \( 23 + 76\cdot 83 + 33\cdot 83^{2} + 16\cdot 83^{3} + 44\cdot 83^{4} +O(83^{5})\) |
$r_{ 3 }$ | $=$ | \( 61 + 6\cdot 83 + 49\cdot 83^{2} + 66\cdot 83^{3} + 38\cdot 83^{4} +O(83^{5})\) |
$r_{ 4 }$ | $=$ | \( 71 + 73\cdot 83^{2} + 52\cdot 83^{3} + 8\cdot 83^{4} +O(83^{5})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |