Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 4.4.4752.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{3}, \sqrt{11})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 36\cdot 83 + 77\cdot 83^{2} + 39\cdot 83^{3} + 7\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 + 80\cdot 83 + 72\cdot 83^{2} + 39\cdot 83^{3} + 30\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 48 + 2\cdot 83 + 10\cdot 83^{2} + 43\cdot 83^{3} + 52\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 75 + 46\cdot 83 + 5\cdot 83^{2} + 43\cdot 83^{3} + 75\cdot 83^{4} +O(83^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |