Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(278784\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.8.21667237072994304.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8$ |
| Parity: | even |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{6}, \sqrt{11})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 20\cdot 97 + 94\cdot 97^{2} + 51\cdot 97^{3} + 43\cdot 97^{4} + 91\cdot 97^{5} + 84\cdot 97^{6} + 67\cdot 97^{7} + 56\cdot 97^{8} + 79\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 37\cdot 97 + 39\cdot 97^{2} + 86\cdot 97^{3} + 13\cdot 97^{4} + 72\cdot 97^{5} + 9\cdot 97^{6} + 11\cdot 97^{7} + 97^{8} + 32\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 37\cdot 97 + 45\cdot 97^{2} + 8\cdot 97^{3} + 30\cdot 97^{4} + 44\cdot 97^{5} + 93\cdot 97^{6} + 67\cdot 97^{7} + 51\cdot 97^{8} + 70\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 32 + 61\cdot 97 + 95\cdot 97^{2} + 39\cdot 97^{3} + 59\cdot 97^{4} + 60\cdot 97^{5} + 39\cdot 97^{7} + 71\cdot 97^{8} + 41\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 65 + 35\cdot 97 + 97^{2} + 57\cdot 97^{3} + 37\cdot 97^{4} + 36\cdot 97^{5} + 96\cdot 97^{6} + 57\cdot 97^{7} + 25\cdot 97^{8} + 55\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 78 + 59\cdot 97 + 51\cdot 97^{2} + 88\cdot 97^{3} + 66\cdot 97^{4} + 52\cdot 97^{5} + 3\cdot 97^{6} + 29\cdot 97^{7} + 45\cdot 97^{8} + 26\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 86 + 59\cdot 97 + 57\cdot 97^{2} + 10\cdot 97^{3} + 83\cdot 97^{4} + 24\cdot 97^{5} + 87\cdot 97^{6} + 85\cdot 97^{7} + 95\cdot 97^{8} + 64\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 88 + 76\cdot 97 + 2\cdot 97^{2} + 45\cdot 97^{3} + 53\cdot 97^{4} + 5\cdot 97^{5} + 12\cdot 97^{6} + 29\cdot 97^{7} + 40\cdot 97^{8} + 17\cdot 97^{9} +O(97^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |