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 8.0.761760000.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - x^{6} - 8x^{5} + 13x^{4} + 4x^{3} + 19x^{2} - 56x + 29 \)
|
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 6\cdot 149 + 91\cdot 149^{2} + 17\cdot 149^{3} + 61\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 58\cdot 149 + 90\cdot 149^{2} + 31\cdot 149^{3} + 6\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 131 + 28\cdot 149 + 81\cdot 149^{2} + 89\cdot 149^{3} + 37\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 137 + 25\cdot 149 + 74\cdot 149^{2} + 55\cdot 149^{3} + 70\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 138 + 54\cdot 149 + 92\cdot 149^{2} + 8\cdot 149^{3} + 97\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 139 + 35\cdot 149 + 52\cdot 149^{2} + 121\cdot 149^{3} + 34\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 144 + 40\cdot 149 + 38\cdot 149^{2} + 135\cdot 149^{3} + 141\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 146 + 46\cdot 149 + 76\cdot 149^{2} + 136\cdot 149^{3} + 146\cdot 149^{4} +O(149^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(2,4)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,4,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,7,4,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,4)(3,5,6,7)$ | $0$ |
| $2$ | $4$ | $(1,5,8,7)(2,3,4,6)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,7,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.