Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Artin stem field: | Galois closure of 8.0.419430400.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{10})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} + 48x^{4} - 60x^{2} + 25 \)
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The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 23\cdot 41 + 3\cdot 41^{2} + 12\cdot 41^{3} + 38\cdot 41^{4} + 10\cdot 41^{5} +O(41^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 5 + 34\cdot 41 + 11\cdot 41^{2} + 37\cdot 41^{3} + 34\cdot 41^{4} + 13\cdot 41^{5} +O(41^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 14 + 41 + 26\cdot 41^{2} + 34\cdot 41^{3} + 18\cdot 41^{4} + 10\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 + 35\cdot 41 + 11\cdot 41^{2} + 8\cdot 41^{3} + 29\cdot 41^{4} + 20\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 + 5\cdot 41 + 29\cdot 41^{2} + 32\cdot 41^{3} + 11\cdot 41^{4} + 20\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 + 39\cdot 41 + 14\cdot 41^{2} + 6\cdot 41^{3} + 22\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 36 + 6\cdot 41 + 29\cdot 41^{2} + 3\cdot 41^{3} + 6\cdot 41^{4} + 27\cdot 41^{5} +O(41^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 39 + 17\cdot 41 + 37\cdot 41^{2} + 28\cdot 41^{3} + 2\cdot 41^{4} + 30\cdot 41^{5} +O(41^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ | |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ | |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |