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 8.0.428490000.2 |
| 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 \(\Q(\sqrt{-15}, \sqrt{-69})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 6x^{6} - 16x^{5} + 40x^{4} - 39x^{3} + 64x^{2} - 98x + 49 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 39\cdot 79 + 3\cdot 79^{2} + 48\cdot 79^{3} + 11\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 36\cdot 79^{2} + 32\cdot 79^{3} + 73\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 26\cdot 79 + 47\cdot 79^{2} + 28\cdot 79^{3} + 22\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 46 + 71\cdot 79 + 52\cdot 79^{2} + 44\cdot 79^{3} + 57\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 50 + 44\cdot 79 + 40\cdot 79^{2} + 50\cdot 79^{3} +O(79^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 55 + 5\cdot 79 + 53\cdot 79^{2} + 28\cdot 79^{3} + 5\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 65 + 29\cdot 79 + 22\cdot 79^{2} + 73\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 78 + 19\cdot 79 + 60\cdot 79^{2} + 9\cdot 79^{3} + 2\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,7)(4,8)$ | $0$ |
| $1$ | $4$ | $(1,8,7,4)(2,6,5,3)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,7,8)(2,3,5,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,7,8)(2,6,5,3)$ | $0$ |
| $2$ | $4$ | $(1,6,7,3)(2,8,5,4)$ | $0$ |
| $2$ | $4$ | $(1,2,7,5)(3,8,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.