Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.8.47775744000000.2 |
| 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 \(\Q(\sqrt{5}, \sqrt{6})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 60x^{6} + 810x^{4} - 1800x^{2} + 900 \)
|
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 140\cdot 149 + 95\cdot 149^{2} + 72\cdot 149^{3} + 116\cdot 149^{4} + 77\cdot 149^{5} + 91\cdot 149^{6} + 45\cdot 149^{7} + 38\cdot 149^{8} + 148\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 28\cdot 149 + 78\cdot 149^{2} + 90\cdot 149^{3} + 40\cdot 149^{4} + 44\cdot 149^{5} + 101\cdot 149^{6} + 135\cdot 149^{7} + 133\cdot 149^{8} + 82\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 + 145\cdot 149 + 22\cdot 149^{2} + 90\cdot 149^{3} + 70\cdot 149^{4} + 47\cdot 149^{5} + 69\cdot 149^{6} + 89\cdot 149^{7} + 46\cdot 149^{8} + 43\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 + 34\cdot 149 + 96\cdot 149^{2} + 89\cdot 149^{3} + 31\cdot 149^{4} + 110\cdot 149^{5} + 34\cdot 149^{6} + 59\cdot 149^{7} + 147\cdot 149^{8} + 21\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 98 + 114\cdot 149 + 52\cdot 149^{2} + 59\cdot 149^{3} + 117\cdot 149^{4} + 38\cdot 149^{5} + 114\cdot 149^{6} + 89\cdot 149^{7} + 149^{8} + 127\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 103 + 3\cdot 149 + 126\cdot 149^{2} + 58\cdot 149^{3} + 78\cdot 149^{4} + 101\cdot 149^{5} + 79\cdot 149^{6} + 59\cdot 149^{7} + 102\cdot 149^{8} + 105\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 109 + 120\cdot 149 + 70\cdot 149^{2} + 58\cdot 149^{3} + 108\cdot 149^{4} + 104\cdot 149^{5} + 47\cdot 149^{6} + 13\cdot 149^{7} + 15\cdot 149^{8} + 66\cdot 149^{9} +O(149^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 148 + 8\cdot 149 + 53\cdot 149^{2} + 76\cdot 149^{3} + 32\cdot 149^{4} + 71\cdot 149^{5} + 57\cdot 149^{6} + 103\cdot 149^{7} + 110\cdot 149^{8} +O(149^{10})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |