Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(624\)\(\medspace = 2^{4} \cdot 3 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.8112.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.156.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{3}, \sqrt{13})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 5x^{2} + 3 \)
|
The roots of $f$ are computed in $\Q_{ 107 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 42\cdot 107 + 107^{2} + 106\cdot 107^{3} + 62\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 45 + 17\cdot 107 + 81\cdot 107^{2} + 6\cdot 107^{3} + 3\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 62 + 89\cdot 107 + 25\cdot 107^{2} + 100\cdot 107^{3} + 103\cdot 107^{4} +O(107^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 86 + 64\cdot 107 + 105\cdot 107^{2} + 44\cdot 107^{4} +O(107^{5})\)
|
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.