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 field: | Galois closure of 8.8.2407470785888256.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{11})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 132x^{6} + 2772x^{4} - 13068x^{2} + 9801 \)
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The roots of $f$ are computed in $\Q_{ 97 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 26\cdot 97 + 68\cdot 97^{2} + 41\cdot 97^{3} + 95\cdot 97^{4} + 9\cdot 97^{5} + 31\cdot 97^{6} + 40\cdot 97^{7} + 62\cdot 97^{8} + 13\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 84\cdot 97 + 68\cdot 97^{2} + 2\cdot 97^{3} + 26\cdot 97^{4} + 57\cdot 97^{5} + 11\cdot 97^{6} + 30\cdot 97^{7} + 88\cdot 97^{8} + 93\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 + 75\cdot 97 + 97^{2} + 16\cdot 97^{3} + 66\cdot 97^{4} + 19\cdot 97^{5} + 29\cdot 97^{6} + 72\cdot 97^{7} + 88\cdot 97^{8} + 80\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 + 91\cdot 97 + 20\cdot 97^{2} + 43\cdot 97^{3} + 96\cdot 97^{4} + 61\cdot 97^{5} + 96\cdot 97^{6} + 80\cdot 97^{7} + 25\cdot 97^{8} + 58\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 55 + 5\cdot 97 + 76\cdot 97^{2} + 53\cdot 97^{3} + 35\cdot 97^{5} + 16\cdot 97^{7} + 71\cdot 97^{8} + 38\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 61 + 21\cdot 97 + 95\cdot 97^{2} + 80\cdot 97^{3} + 30\cdot 97^{4} + 77\cdot 97^{5} + 67\cdot 97^{6} + 24\cdot 97^{7} + 8\cdot 97^{8} + 16\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 65 + 12\cdot 97 + 28\cdot 97^{2} + 94\cdot 97^{3} + 70\cdot 97^{4} + 39\cdot 97^{5} + 85\cdot 97^{6} + 66\cdot 97^{7} + 8\cdot 97^{8} + 3\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 92 + 70\cdot 97 + 28\cdot 97^{2} + 55\cdot 97^{3} + 97^{4} + 87\cdot 97^{5} + 65\cdot 97^{6} + 56\cdot 97^{7} + 34\cdot 97^{8} + 83\cdot 97^{9} +O(97^{10})\)
|
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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.