Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(127449\)\(\medspace = 3^{2} \cdot 7^{2} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.8.2070185663499849.1 |
| 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 \(\Q(\sqrt{17}, \sqrt{21})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} - 110x^{6} + 153x^{5} + 3789x^{4} + 1989x^{3} - 44000x^{2} - 97899x - 46703 \)
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The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 44\cdot 47 + 27\cdot 47^{2} + 36\cdot 47^{3} + 21\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 6\cdot 47^{3} + 36\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 25\cdot 47 + 25\cdot 47^{2} + 14\cdot 47^{3} + 44\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 2\cdot 47 + 30\cdot 47^{2} + 27\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 43\cdot 47 + 34\cdot 47^{2} + 11\cdot 47^{3} + 30\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 + 3\cdot 47 + 38\cdot 47^{2} + 10\cdot 47^{3} + 16\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 + 42\cdot 47 + 25\cdot 47^{2} + 7\cdot 47^{3} + 8\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 44 + 25\cdot 47 + 5\cdot 47^{2} + 6\cdot 47^{3} + 4\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $-2$ | |
| $2$ | $4$ | $(1,3,8,2)(4,5,6,7)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,5)(2,4,3,6)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,4)(2,7,3,5)$ | $0$ |