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 8.0.428490000.1 |
| 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 \(\Q(\sqrt{-15}, \sqrt{-69})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 11x^{6} - 11x^{5} + 30x^{4} + x^{3} + 39x^{2} + 37x + 19 \)
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The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 3\cdot 19^{2} + 15\cdot 19^{3} + 6\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 18\cdot 19 + 13\cdot 19^{2} + 18\cdot 19^{3} + 17\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 5\cdot 19 + 9\cdot 19^{2} + 9\cdot 19^{3} + 13\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 + 3\cdot 19 + 5\cdot 19^{2} + 11\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 + 19 + 6\cdot 19^{2} + 16\cdot 19^{3} + 14\cdot 19^{4} +O(19^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 9 + 15\cdot 19 + 18\cdot 19^{2} + 7\cdot 19^{3} + 18\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 + 11\cdot 19 + 19^{2} + 11\cdot 19^{3} + 13\cdot 19^{4} +O(19^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 18 + 18\cdot 19^{2} + 4\cdot 19^{3} + 17\cdot 19^{4} +O(19^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,5,8,2)(3,4,6,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,5)(3,7,6,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,7)(2,3,5,6)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,8,2)(3,7,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.