Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Artin stem field: | Galois closure of 8.0.9032601600.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.440.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-6}, \sqrt{-55})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 2x^{6} - 10x^{5} + 112x^{4} - 272x^{3} + 341x^{2} - 434x + 343 \)
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The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 181\cdot 337 + 331\cdot 337^{2} + 143\cdot 337^{3} + 234\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 79 + 189\cdot 337 + 54\cdot 337^{2} + 15\cdot 337^{3} + 331\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 102 + 305\cdot 337 + 130\cdot 337^{2} + 210\cdot 337^{3} + 161\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 141 + 162\cdot 337 + 67\cdot 337^{2} + 165\cdot 337^{3} + 85\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 194 + 238\cdot 337 + 86\cdot 337^{2} + 118\cdot 337^{3} + 224\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 213 + 317\cdot 337 + 294\cdot 337^{2} + 61\cdot 337^{3} + 190\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 300 + 12\cdot 337 + 317\cdot 337^{2} + 302\cdot 337^{3} + 163\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 301 + 277\cdot 337 + 64\cdot 337^{2} + 330\cdot 337^{3} + 293\cdot 337^{4} +O(337^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $-2$ | |
| $2$ | $2$ | $(2,8)(4,6)$ | $0$ | |
| $2$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | |
| $1$ | $4$ | $(1,3,7,5)(2,6,8,4)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,5,7,3)(2,4,8,6)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,3,7,5)(2,4,8,6)$ | $0$ | |
| $2$ | $4$ | $(1,8,7,2)(3,4,5,6)$ | $0$ | |
| $2$ | $4$ | $(1,4,7,6)(2,5,8,3)$ | $0$ |