Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.8.12230590464.1 |
| 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 \(\Q(\sqrt{2}, \sqrt{3})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} + 36x^{4} - 36x^{2} + 9 \)
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The roots of $f$ are computed in $\Q_{ 23 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 8\cdot 23 + 20\cdot 23^{2} + 8\cdot 23^{3} + 16\cdot 23^{4} + 17\cdot 23^{5} + 20\cdot 23^{6} + 15\cdot 23^{7} + 19\cdot 23^{8} + 6\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 19\cdot 23 + 16\cdot 23^{2} + 4\cdot 23^{3} + 16\cdot 23^{4} + 14\cdot 23^{5} + 19\cdot 23^{6} + 3\cdot 23^{7} + 20\cdot 23^{8} + 22\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 8\cdot 23 + 22\cdot 23^{2} + 17\cdot 23^{3} + 19\cdot 23^{4} + 5\cdot 23^{5} + 9\cdot 23^{6} + 15\cdot 23^{7} + 14\cdot 23^{8} + 17\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 11\cdot 23 + 21\cdot 23^{2} + 10\cdot 23^{3} + 6\cdot 23^{4} + 20\cdot 23^{5} + 17\cdot 23^{6} + 3\cdot 23^{7} + 14\cdot 23^{8} + 19\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 11\cdot 23 + 23^{2} + 12\cdot 23^{3} + 16\cdot 23^{4} + 2\cdot 23^{5} + 5\cdot 23^{6} + 19\cdot 23^{7} + 8\cdot 23^{8} + 3\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 + 14\cdot 23 + 5\cdot 23^{3} + 3\cdot 23^{4} + 17\cdot 23^{5} + 13\cdot 23^{6} + 7\cdot 23^{7} + 8\cdot 23^{8} + 5\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 14 + 3\cdot 23 + 6\cdot 23^{2} + 18\cdot 23^{3} + 6\cdot 23^{4} + 8\cdot 23^{5} + 3\cdot 23^{6} + 19\cdot 23^{7} + 2\cdot 23^{8} +O(23^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 20 + 14\cdot 23 + 2\cdot 23^{2} + 14\cdot 23^{3} + 6\cdot 23^{4} + 5\cdot 23^{5} + 2\cdot 23^{6} + 7\cdot 23^{7} + 3\cdot 23^{8} + 16\cdot 23^{9} +O(23^{10})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |