Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(95\)\(\medspace = 5 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.475.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.95.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-19})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{3} + 2x^{2} - x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 + 56\cdot 131 + 56\cdot 131^{2} + 104\cdot 131^{3} +O(131^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 40 + 47\cdot 131 + 85\cdot 131^{2} + 72\cdot 131^{3} + 46\cdot 131^{4} +O(131^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 92 + 83\cdot 131 + 45\cdot 131^{2} + 58\cdot 131^{3} + 84\cdot 131^{4} +O(131^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 97 + 74\cdot 131 + 74\cdot 131^{2} + 26\cdot 131^{3} + 130\cdot 131^{4} +O(131^{5})\)
| |
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.