Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(189225\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 29^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.752823265640625.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{29})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 98x^{6} - 105x^{5} + 3191x^{4} + 1665x^{3} + 44072x^{2} + 47933x + 328171 \)
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The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 12\cdot 109 + 75\cdot 109^{2} + 81\cdot 109^{3} + 86\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 89\cdot 109 + 51\cdot 109^{2} + 53\cdot 109^{3} + 57\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 62\cdot 109 + 77\cdot 109^{2} + 45\cdot 109^{3} + 53\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 + 17\cdot 109 + 10\cdot 109^{2} + 100\cdot 109^{3} + 106\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 63 + 21\cdot 109 + 20\cdot 109^{2} + 31\cdot 109^{3} + 37\cdot 109^{4} +O(109^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 92 + 90\cdot 109 + 103\cdot 109^{2} + 82\cdot 109^{3} + 101\cdot 109^{4} +O(109^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 94 + 20\cdot 109 + 66\cdot 109^{2} + 94\cdot 109^{4} +O(109^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 102 + 12\cdot 109 + 31\cdot 109^{2} + 40\cdot 109^{3} + 7\cdot 109^{4} +O(109^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,3,2,4)(5,8,6,7)$ | $0$ | |
| $2$ | $4$ | $(1,8,2,7)(3,5,4,6)$ | $0$ | |
| $2$ | $4$ | $(1,5,2,6)(3,7,4,8)$ | $0$ |