Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(112896\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.1438916737499136.2 |
| 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 \(\Q(\sqrt{3}, \sqrt{14})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 84x^{6} + 2268x^{4} + 19404x^{2} + 441 \)
|
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 34 + 17\cdot 157 + 56\cdot 157^{2} + 65\cdot 157^{3} + 131\cdot 157^{4} + 142\cdot 157^{5} + 85\cdot 157^{6} + 4\cdot 157^{7} + 35\cdot 157^{8} +O(157^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 + 41\cdot 157 + 13\cdot 157^{2} + 8\cdot 157^{3} + 31\cdot 157^{4} + 131\cdot 157^{5} + 47\cdot 157^{6} + 95\cdot 157^{7} + 97\cdot 157^{8} +O(157^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 67 + 2\cdot 157 + 144\cdot 157^{2} + 58\cdot 157^{3} + 55\cdot 157^{4} + 11\cdot 157^{5} + 8\cdot 157^{6} + 147\cdot 157^{7} + 30\cdot 157^{8} +O(157^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 76 + 123\cdot 157 + 68\cdot 157^{2} + 52\cdot 157^{3} + 118\cdot 157^{4} + 29\cdot 157^{5} + 67\cdot 157^{6} + 93\cdot 157^{7} + 138\cdot 157^{8} +O(157^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 81 + 33\cdot 157 + 88\cdot 157^{2} + 104\cdot 157^{3} + 38\cdot 157^{4} + 127\cdot 157^{5} + 89\cdot 157^{6} + 63\cdot 157^{7} + 18\cdot 157^{8} +O(157^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 90 + 154\cdot 157 + 12\cdot 157^{2} + 98\cdot 157^{3} + 101\cdot 157^{4} + 145\cdot 157^{5} + 148\cdot 157^{6} + 9\cdot 157^{7} + 126\cdot 157^{8} +O(157^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 120 + 115\cdot 157 + 143\cdot 157^{2} + 148\cdot 157^{3} + 125\cdot 157^{4} + 25\cdot 157^{5} + 109\cdot 157^{6} + 61\cdot 157^{7} + 59\cdot 157^{8} +O(157^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 123 + 139\cdot 157 + 100\cdot 157^{2} + 91\cdot 157^{3} + 25\cdot 157^{4} + 14\cdot 157^{5} + 71\cdot 157^{6} + 152\cdot 157^{7} + 121\cdot 157^{8} +O(157^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |