Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(198025\)\(\medspace = 5^{2} \cdot 89^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.7765332671265625.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{89})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 78x^{6} - 100x^{5} + 2541x^{4} - 7670x^{3} + 65772x^{2} - 36712x + 1485296 \)
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The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 81\cdot 139 + 91\cdot 139^{2} + 24\cdot 139^{3} + 117\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 47 + 71\cdot 139 + 15\cdot 139^{2} + 39\cdot 139^{3} + 128\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 65 + 102\cdot 139 + 123\cdot 139^{2} + 35\cdot 139^{3} + 74\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 82 + 8\cdot 139 + 18\cdot 139^{2} + 108\cdot 139^{3} + 100\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 107 + 49\cdot 139 + 102\cdot 139^{2} + 13\cdot 139^{3} + 64\cdot 139^{4} +O(139^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 120 + 108\cdot 139 + 53\cdot 139^{2} + 91\cdot 139^{3} + 91\cdot 139^{4} +O(139^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 126 + 37\cdot 139 + 83\cdot 139^{2} + 3\cdot 139^{3} + 7\cdot 139^{4} +O(139^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 128 + 95\cdot 139 + 67\cdot 139^{2} + 100\cdot 139^{3} + 111\cdot 139^{4} +O(139^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,7,3,5)(2,8,4,6)$ | $0$ | |
| $2$ | $4$ | $(1,4,3,2)(5,6,7,8)$ | $0$ | |
| $2$ | $4$ | $(1,8,3,6)(2,5,4,7)$ | $0$ |