Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(392\)\(\medspace = 2^{3} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.481890304.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{-7})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 11\cdot 67 + 14\cdot 67^{2} + 46\cdot 67^{3} + 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 45\cdot 67 + 12\cdot 67^{2} + 9\cdot 67^{3} + 60\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 53\cdot 67 + 40\cdot 67^{2} + 52\cdot 67^{3} + 52\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 60\cdot 67 + 14\cdot 67^{2} + 25\cdot 67^{3} + 18\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 + 58\cdot 67 + 21\cdot 67^{2} + 66\cdot 67^{3} + 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 + 9\cdot 67 + 64\cdot 67^{2} + 9\cdot 67^{3} + 61\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 50 + 5\cdot 67 + 33\cdot 67^{2} + 32\cdot 67^{3} + 52\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 53 + 24\cdot 67 + 66\cdot 67^{2} + 25\cdot 67^{3} + 19\cdot 67^{4} +O(67^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,6,5,8)(2,3,4,7)$ | $0$ |