Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(119025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.8.1686221298140625.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{69})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 112x^{6} + 95x^{5} + 2881x^{4} + 835x^{3} - 16858x^{2} - 25817x - 10769 \)
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The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 59\cdot 89 + 83\cdot 89^{2} + 7\cdot 89^{3} + 74\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 17\cdot 89^{2} + 41\cdot 89^{3} + 22\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 68\cdot 89 + 66\cdot 89^{2} + 22\cdot 89^{3} + 59\cdot 89^{4} +O(89^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 46 + 36\cdot 89 + 72\cdot 89^{2} + 19\cdot 89^{4} +O(89^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 51 + 79\cdot 89 + 36\cdot 89^{2} + 81\cdot 89^{3} + 87\cdot 89^{4} +O(89^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 68 + 52\cdot 89 + 70\cdot 89^{2} + 52\cdot 89^{3} + 54\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 78 + 32\cdot 89 + 11\cdot 89^{2} + 52\cdot 89^{3} + 16\cdot 89^{4} +O(89^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 80 + 26\cdot 89 + 86\cdot 89^{2} + 7\cdot 89^{3} + 22\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,2)(3,4)(5,8)(6,7)$ | $-2$ | |
| $2$ | $4$ | $(1,7,2,6)(3,5,4,8)$ | $0$ | |
| $2$ | $4$ | $(1,5,2,8)(3,6,4,7)$ | $0$ | |
| $2$ | $4$ | $(1,3,2,4)(5,7,8,6)$ | $0$ |