Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1380\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 23 \) |
| Artin stem field: | Galois closure of 8.0.761760000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.1380.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{-69})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 5x^{6} + 8x^{5} - 9x^{4} + 20x^{3} + 9x^{2} - 16x + 29 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 72\cdot 89 + 51\cdot 89^{2} + 87\cdot 89^{3} + 5\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 46\cdot 89 + 53\cdot 89^{2} + 45\cdot 89^{3} + 45\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 48 + 58\cdot 89 + 29\cdot 89^{2} + 5\cdot 89^{3} + 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 58 + 19\cdot 89 + 54\cdot 89^{2} + 89^{3} + 17\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 76 + 68\cdot 89 + 30\cdot 89^{2} + 6\cdot 89^{3} + 81\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 77 + 11\cdot 89 + 53\cdot 89^{2} + 10\cdot 89^{3} + 14\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 78 + 79\cdot 89 + 41\cdot 89^{2} + 38\cdot 89^{3} + 45\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 85 + 87\cdot 89 + 40\cdot 89^{2} + 71\cdot 89^{3} + 56\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $-2$ |
| $2$ | $2$ | $(3,8)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,7,2,5)(3,4,8,6)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,2,7)(3,6,8,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,2,6)(3,7,8,5)$ | $0$ |
| $2$ | $4$ | $(1,7,2,5)(3,6,8,4)$ | $0$ |
| $2$ | $4$ | $(1,3,2,8)(4,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.