Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Artin stem field: | Galois closure of 8.0.70560000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.420.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{-21})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + x^{6} + 4x^{5} + 3x^{4} + 8x^{3} + 9x^{2} + 4x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 58\cdot 89 + 9\cdot 89^{2} + 69\cdot 89^{3} + 53\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 60\cdot 89 + 6\cdot 89^{2} + 64\cdot 89^{3} + 86\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 83\cdot 89 + 87\cdot 89^{2} + 6\cdot 89^{3} + 31\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 + 65\cdot 89 + 73\cdot 89^{2} + 37\cdot 89^{3} + 6\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 59 + 63\cdot 89 + 50\cdot 89^{2} + 52\cdot 89^{3} + 78\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 + 30\cdot 89 + 70\cdot 89^{2} + 9\cdot 89^{3} + 77\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 69 + 26\cdot 89 + 67\cdot 89^{3} + 69\cdot 89^{4} +O(89^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 76 + 56\cdot 89 + 56\cdot 89^{2} + 48\cdot 89^{3} + 41\cdot 89^{4} +O(89^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $2$ | $2$ | $(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,3,2,4)(5,7,8,6)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,3)(5,6,8,7)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,2,5)(3,6,4,7)$ | $0$ |
| $2$ | $4$ | $(1,7,2,6)(3,8,4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.