Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(407\)\(\medspace = 11 \cdot 37 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.4477.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.407.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-11}, \sqrt{37})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 4x + 5 \) . |
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 36\cdot 47 + 39\cdot 47^{2} + 16\cdot 47^{3} + 16\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 24 + 32\cdot 47 + 16\cdot 47^{3} + 6\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 31 + 25\cdot 47 + 18\cdot 47^{2} + 29\cdot 47^{3} + 45\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 37 + 46\cdot 47 + 34\cdot 47^{2} + 31\cdot 47^{3} + 25\cdot 47^{4} +O(47^{5})\) |
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 value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$2$ | $2$ | $(1,3)(2,4)$ | $0$ |
$2$ | $2$ | $(1,2)$ | $0$ |
$2$ | $4$ | $(1,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.