Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(495\)\(\medspace = 3^{2} \cdot 5 \cdot 11 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.5445.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.55.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 4x^{2} - 6x + 3 \) . |
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 59 + 66\cdot 89 + 74\cdot 89^{2} + 72\cdot 89^{3} + 39\cdot 89^{4} +O(89^{5})\) |
$r_{ 2 }$ | $=$ | \( 67 + 73\cdot 89 + 29\cdot 89^{2} + 3\cdot 89^{3} + 14\cdot 89^{4} +O(89^{5})\) |
$r_{ 3 }$ | $=$ | \( 69 + 64\cdot 89 + 67\cdot 89^{2} + 48\cdot 89^{3} + 62\cdot 89^{4} +O(89^{5})\) |
$r_{ 4 }$ | $=$ | \( 73 + 61\cdot 89 + 5\cdot 89^{2} + 53\cdot 89^{3} + 61\cdot 89^{4} +O(89^{5})\) |
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.