Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(92416\)\(\medspace = 2^{8} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.789298907447296.1 |
| 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 \(\Q(\sqrt{2}, \sqrt{19})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 76x^{6} + 1748x^{4} + 12996x^{2} + 29241 \)
|
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 12\cdot 73 + 32\cdot 73^{2} + 22\cdot 73^{3} + 35\cdot 73^{4} + 25\cdot 73^{5} + 51\cdot 73^{6} + 62\cdot 73^{7} + 34\cdot 73^{8} + 63\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 15\cdot 73 + 65\cdot 73^{2} + 9\cdot 73^{3} + 40\cdot 73^{4} + 56\cdot 73^{5} + 11\cdot 73^{6} + 2\cdot 73^{7} + 7\cdot 73^{8} + 3\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 34\cdot 73 + 72\cdot 73^{2} + 40\cdot 73^{3} + 42\cdot 73^{4} + 40\cdot 73^{5} + 39\cdot 73^{6} + 37\cdot 73^{7} + 46\cdot 73^{8} + 8\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 46\cdot 73 + 17\cdot 73^{2} + 7\cdot 73^{3} + 41\cdot 73^{4} + 50\cdot 73^{5} + 34\cdot 73^{6} + 46\cdot 73^{7} + 70\cdot 73^{8} + 52\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 44 + 26\cdot 73 + 55\cdot 73^{2} + 65\cdot 73^{3} + 31\cdot 73^{4} + 22\cdot 73^{5} + 38\cdot 73^{6} + 26\cdot 73^{7} + 2\cdot 73^{8} + 20\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 48 + 38\cdot 73 + 32\cdot 73^{3} + 30\cdot 73^{4} + 32\cdot 73^{5} + 33\cdot 73^{6} + 35\cdot 73^{7} + 26\cdot 73^{8} + 64\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 50 + 57\cdot 73 + 7\cdot 73^{2} + 63\cdot 73^{3} + 32\cdot 73^{4} + 16\cdot 73^{5} + 61\cdot 73^{6} + 70\cdot 73^{7} + 65\cdot 73^{8} + 69\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 51 + 60\cdot 73 + 40\cdot 73^{2} + 50\cdot 73^{3} + 37\cdot 73^{4} + 47\cdot 73^{5} + 21\cdot 73^{6} + 10\cdot 73^{7} + 38\cdot 73^{8} + 9\cdot 73^{9} +O(73^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |