Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(53361\)\(\medspace = 3^{2} \cdot 7^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.151939915084881.1 |
| 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 \(\Q(\sqrt{21}, \sqrt{33})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 50x^{6} + 71x^{5} + 529x^{4} + 2173x^{3} + 842x^{2} + 5545x + 17623 \)
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The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 22\cdot 67 + 21\cdot 67^{2} + 22\cdot 67^{3} + 56\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 4\cdot 67 + 3\cdot 67^{2} + 14\cdot 67^{3} + 4\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 49\cdot 67 + 36\cdot 67^{2} + 12\cdot 67^{3} + 38\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 14\cdot 67 + 51\cdot 67^{2} + 19\cdot 67^{3} + 51\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 30\cdot 67 + 51\cdot 67^{2} + 62\cdot 67^{3} + 17\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 + 25\cdot 67 + 41\cdot 67^{2} + 51\cdot 67^{3} + 22\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 38 + 33\cdot 67 + 7\cdot 67^{2} + 41\cdot 67^{3} + 58\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 54 + 21\cdot 67 + 55\cdot 67^{2} + 43\cdot 67^{3} + 18\cdot 67^{4} +O(67^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ |
| $2$ | $4$ | $(1,6,4,7)(2,8,5,3)$ | $0$ |
| $2$ | $4$ | $(1,3,4,8)(2,6,5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,4,2)(3,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.