Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.339738624.6 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 13\cdot 97 + 50\cdot 97^{2} + 89\cdot 97^{3} + 64\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 20\cdot 97 + 47\cdot 97^{2} + 68\cdot 97^{3} + 67\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 50\cdot 97 + 55\cdot 97^{2} + 32\cdot 97^{3} + 48\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 + 57\cdot 97 + 52\cdot 97^{2} + 11\cdot 97^{3} + 51\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 63 + 80\cdot 97 + 16\cdot 97^{2} + 72\cdot 97^{3} + 57\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 69 + 5\cdot 97 + 69\cdot 97^{2} + 77\cdot 97^{3} + 36\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 80 + 20\cdot 97 + 22\cdot 97^{2} + 15\cdot 97^{3} + 41\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 86 + 42\cdot 97 + 74\cdot 97^{2} + 20\cdot 97^{3} + 20\cdot 97^{4} +O(97^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,4)(2,3,6,7)$ | $0$ |