Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(73984\)\(\medspace = 2^{8} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.8.101240302206976.2 |
| 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 \(\Q(\sqrt{2}, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 68x^{6} + 986x^{4} - 4624x^{2} + 4624 \)
|
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 15\cdot 137 + 21\cdot 137^{2} + 110\cdot 137^{3} + 90\cdot 137^{4} + 107\cdot 137^{5} + 129\cdot 137^{6} + 25\cdot 137^{7} + 64\cdot 137^{8} + 103\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 71\cdot 137 + 26\cdot 137^{2} + 137^{3} + 82\cdot 137^{4} + 23\cdot 137^{5} + 17\cdot 137^{6} + 50\cdot 137^{7} + 131\cdot 137^{8} + 37\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 + 121\cdot 137 + 22\cdot 137^{2} + 28\cdot 137^{3} + 44\cdot 137^{4} + 95\cdot 137^{5} + 66\cdot 137^{6} + 103\cdot 137^{7} + 55\cdot 137^{8} + 65\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 60 + 136\cdot 137 + 94\cdot 137^{2} + 99\cdot 137^{3} + 84\cdot 137^{4} + 63\cdot 137^{5} + 52\cdot 137^{6} + 70\cdot 137^{7} + 50\cdot 137^{8} + 33\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 77 + 42\cdot 137^{2} + 37\cdot 137^{3} + 52\cdot 137^{4} + 73\cdot 137^{5} + 84\cdot 137^{6} + 66\cdot 137^{7} + 86\cdot 137^{8} + 103\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 90 + 15\cdot 137 + 114\cdot 137^{2} + 108\cdot 137^{3} + 92\cdot 137^{4} + 41\cdot 137^{5} + 70\cdot 137^{6} + 33\cdot 137^{7} + 81\cdot 137^{8} + 71\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 134 + 65\cdot 137 + 110\cdot 137^{2} + 135\cdot 137^{3} + 54\cdot 137^{4} + 113\cdot 137^{5} + 119\cdot 137^{6} + 86\cdot 137^{7} + 5\cdot 137^{8} + 99\cdot 137^{9} +O(137^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 135 + 121\cdot 137 + 115\cdot 137^{2} + 26\cdot 137^{3} + 46\cdot 137^{4} + 29\cdot 137^{5} + 7\cdot 137^{6} + 111\cdot 137^{7} + 72\cdot 137^{8} + 33\cdot 137^{9} +O(137^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |