Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(221\)\(\medspace = 13 \cdot 17 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.2873.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Determinant: | 1.221.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{13}, \sqrt{17})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + x^{2} - 2x + 4 \) . |
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 8 + 77\cdot 101 + 41\cdot 101^{2} + 79\cdot 101^{3} + 32\cdot 101^{4} +O(101^{5})\) |
$r_{ 2 }$ | $=$ | \( 42 + 91\cdot 101 + 31\cdot 101^{2} + 19\cdot 101^{3} + 15\cdot 101^{4} +O(101^{5})\) |
$r_{ 3 }$ | $=$ | \( 76 + 83\cdot 101^{2} + 24\cdot 101^{3} + 34\cdot 101^{4} +O(101^{5})\) |
$r_{ 4 }$ | $=$ | \( 77 + 32\cdot 101 + 45\cdot 101^{2} + 78\cdot 101^{3} + 18\cdot 101^{4} +O(101^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,3)$ | $0$ |
$2$ | $4$ | $(1,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.