Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(93025\)\(\medspace = 5^{2} \cdot 61^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.805005849390625.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{61})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 57x^{6} + 135x^{5} + 306x^{4} + 5365x^{3} + 23383x^{2} + 40951x + 75421 \)
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The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 106\cdot 199 + 78\cdot 199^{2} + 61\cdot 199^{3} + 63\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 25\cdot 199 + 148\cdot 199^{2} + 111\cdot 199^{3} + 58\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 179\cdot 199 + 8\cdot 199^{2} + 14\cdot 199^{3} + 187\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 + 41\cdot 199 + 63\cdot 199^{2} + 50\cdot 199^{3} + 23\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 79 + 131\cdot 199 + 171\cdot 199^{2} + 154\cdot 199^{3} + 177\cdot 199^{4} +O(199^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 118 + 37\cdot 199 + 58\cdot 199^{2} + 80\cdot 199^{3} + 74\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 157 + 72\cdot 199 + 156\cdot 199^{2} + 13\cdot 199^{3} + 121\cdot 199^{4} +O(199^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 195 + 3\cdot 199 + 111\cdot 199^{2} + 110\cdot 199^{3} + 90\cdot 199^{4} +O(199^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,6,3,2)(4,7,8,5)$ | $0$ | |
| $2$ | $4$ | $(1,8,3,4)(2,5,6,7)$ | $0$ | |
| $2$ | $4$ | $(1,7,3,5)(2,8,6,4)$ | $0$ |