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 8.0.2070185663499849.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 65x^{6} - 439x^{5} + 1876x^{4} - 12191x^{3} + 60887x^{2} - 124718x + 121291 \)
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The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 + 59\cdot 89 + 69\cdot 89^{2} + 71\cdot 89^{3} + 27\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 75\cdot 89 + 57\cdot 89^{2} + 78\cdot 89^{3} + 11\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 65\cdot 89 + 57\cdot 89^{2} + 13\cdot 89^{3} + 44\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 66\cdot 89 + 82\cdot 89^{2} + 11\cdot 89^{3} + 10\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 + 24\cdot 89 + 11\cdot 89^{2} + 31\cdot 89^{3} + 12\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 67 + 82\cdot 89 + 53\cdot 89^{2} + 29\cdot 89^{3} + 20\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 82 + 66\cdot 89 + 31\cdot 89^{2} + 46\cdot 89^{3} + 19\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 86 + 4\cdot 89 + 80\cdot 89^{2} + 72\cdot 89^{3} + 31\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $0$ | |
| $2$ | $4$ | $(1,6,4,7)(2,5,3,8)$ | $0$ | |
| $2$ | $4$ | $(1,8,4,5)(2,6,3,7)$ | $0$ |