Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(468\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.31539456.5 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.52.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{39})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - x^{6} - 2x^{5} + 10x^{4} + 8x^{3} - 22x^{2} - 4x + 13 \)
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The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 4\cdot 61 + 41\cdot 61^{2} + 9\cdot 61^{3} + 14\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 3\cdot 61 + 57\cdot 61^{2} + 16\cdot 61^{3} + 17\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 29\cdot 61 + 36\cdot 61^{2} + 38\cdot 61^{3} + 17\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 19\cdot 61 + 58\cdot 61^{2} + 49\cdot 61^{3} + 59\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 34\cdot 61 + 26\cdot 61^{2} + 45\cdot 61^{3} + 30\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 + 10\cdot 61 + 6\cdot 61^{2} + 21\cdot 61^{3} + 41\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 + 34\cdot 61 + 20\cdot 61^{2} + 40\cdot 61^{3} + 56\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 45 + 47\cdot 61 + 58\cdot 61^{2} + 21\cdot 61^{3} + 6\cdot 61^{4} +O(61^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,4)(3,8)(6,7)$ | $-2$ | |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ | |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(2,4)(3,8)$ | $0$ | |
| $1$ | $4$ | $(1,6,5,7)(2,3,4,8)$ | $-2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,7,5,6)(2,8,4,3)$ | $2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,8,5,3)(2,7,4,6)$ | $0$ | |
| $2$ | $4$ | $(1,6,5,7)(2,8,4,3)$ | $0$ | |
| $2$ | $4$ | $(1,4,5,2)(3,6,8,7)$ | $0$ |