Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.4.4400.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | even |
Determinant: | 1.220.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{11})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - 7x^{2} + 11 \) . |
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 43 + 86\cdot 131 + 88\cdot 131^{2} + 100\cdot 131^{3} + 114\cdot 131^{4} +O(131^{5})\)
$r_{ 2 }$ |
$=$ |
\( 56 + 81\cdot 131 + 126\cdot 131^{2} + 42\cdot 131^{3} + 112\cdot 131^{4} +O(131^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 75 + 49\cdot 131 + 4\cdot 131^{2} + 88\cdot 131^{3} + 18\cdot 131^{4} +O(131^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 88 + 44\cdot 131 + 42\cdot 131^{2} + 30\cdot 131^{3} + 16\cdot 131^{4} +O(131^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ |
$2$ | $2$ | $(1,4)$ | $0$ |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.