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 8.0.2407470785888256.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8$ |
| Parity: | even |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{11})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 74\cdot 79 + 7\cdot 79^{2} + 24\cdot 79^{3} + 60\cdot 79^{4} + 77\cdot 79^{5} + 62\cdot 79^{6} + 23\cdot 79^{7} + 49\cdot 79^{8} + 30\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 53\cdot 79 + 71\cdot 79^{2} + 19\cdot 79^{3} + 42\cdot 79^{4} + 26\cdot 79^{5} + 75\cdot 79^{6} + 69\cdot 79^{7} + 66\cdot 79^{8} + 46\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 + 66\cdot 79 + 62\cdot 79^{2} + 53\cdot 79^{3} + 61\cdot 79^{4} + 7\cdot 79^{5} + 79^{6} + 54\cdot 79^{7} + 49\cdot 79^{8} + 46\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 + 52\cdot 79 + 58\cdot 79^{2} + 33\cdot 79^{3} + 32\cdot 79^{4} + 67\cdot 79^{5} + 37\cdot 79^{6} + 43\cdot 79^{7} + 34\cdot 79^{8} + 6\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 41 + 26\cdot 79 + 20\cdot 79^{2} + 45\cdot 79^{3} + 46\cdot 79^{4} + 11\cdot 79^{5} + 41\cdot 79^{6} + 35\cdot 79^{7} + 44\cdot 79^{8} + 72\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 12\cdot 79 + 16\cdot 79^{2} + 25\cdot 79^{3} + 17\cdot 79^{4} + 71\cdot 79^{5} + 77\cdot 79^{6} + 24\cdot 79^{7} + 29\cdot 79^{8} + 32\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 53 + 25\cdot 79 + 7\cdot 79^{2} + 59\cdot 79^{3} + 36\cdot 79^{4} + 52\cdot 79^{5} + 3\cdot 79^{6} + 9\cdot 79^{7} + 12\cdot 79^{8} + 32\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 64 + 4\cdot 79 + 71\cdot 79^{2} + 54\cdot 79^{3} + 18\cdot 79^{4} + 79^{5} + 16\cdot 79^{6} + 55\cdot 79^{7} + 29\cdot 79^{8} + 48\cdot 79^{9} +O(79^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |