Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.0.2916000000.5 |
| 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 \(\Q(\zeta_{12})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 15x^{6} + 90x^{4} - 225x^{2} + 225 \)
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The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 30\cdot 61 + 7\cdot 61^{2} + 56\cdot 61^{3} + 3\cdot 61^{4} + 33\cdot 61^{5} +O(61^{6})\)
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| $r_{ 2 }$ | $=$ |
\( 18 + 59\cdot 61 + 46\cdot 61^{2} + 20\cdot 61^{3} + 34\cdot 61^{4} + 38\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 10\cdot 61 + 50\cdot 61^{2} + 58\cdot 61^{3} + 19\cdot 61^{4} + 22\cdot 61^{5} +O(61^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 27 + 47\cdot 61 + 42\cdot 61^{2} + 26\cdot 61^{3} + 22\cdot 61^{4} + 54\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 + 13\cdot 61 + 18\cdot 61^{2} + 34\cdot 61^{3} + 38\cdot 61^{4} + 6\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 + 50\cdot 61 + 10\cdot 61^{2} + 2\cdot 61^{3} + 41\cdot 61^{4} + 38\cdot 61^{5} +O(61^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 43 + 61 + 14\cdot 61^{2} + 40\cdot 61^{3} + 26\cdot 61^{4} + 22\cdot 61^{5} +O(61^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 59 + 30\cdot 61 + 53\cdot 61^{2} + 4\cdot 61^{3} + 57\cdot 61^{4} + 27\cdot 61^{5} +O(61^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ | |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ | |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ | ✓ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |