Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| Artin stem field: | Galois closure of 8.4.33973862400.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.15.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{6})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} - 28x^{4} - 100x^{2} + 25 \)
|
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 34 + 25\cdot 167 + 32\cdot 167^{2} + 116\cdot 167^{3} + 146\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 58 + 2\cdot 167 + 156\cdot 167^{2} + 110\cdot 167^{3} + 70\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 76 + 99\cdot 167 + 42\cdot 167^{2} + 14\cdot 167^{3} + 135\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 80 + 102\cdot 167 + 109\cdot 167^{2} + 95\cdot 167^{3} + 121\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 87 + 64\cdot 167 + 57\cdot 167^{2} + 71\cdot 167^{3} + 45\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 91 + 67\cdot 167 + 124\cdot 167^{2} + 152\cdot 167^{3} + 31\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 109 + 164\cdot 167 + 10\cdot 167^{2} + 56\cdot 167^{3} + 96\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 133 + 141\cdot 167 + 134\cdot 167^{2} + 50\cdot 167^{3} + 20\cdot 167^{4} +O(167^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.