Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:C_2^2$ |
| Conductor: | \(1254400\)\(\medspace = 2^{10} \cdot 5^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.321126400.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8:C_2^2$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^4$ |
| Projective field: | Galois closure of 16.0.9671731157401600000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} - 2x^{6} + 16x^{5} + 8x^{4} - 28x^{3} - 16x^{2} + 24x + 18 \)
|
The roots of $f$ are computed in $\Q_{ 569 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 379\cdot 569 + 346\cdot 569^{2} + 168\cdot 569^{3} + 184\cdot 569^{4} +O(569^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 188\cdot 569 + 534\cdot 569^{2} + 123\cdot 569^{3} + 356\cdot 569^{4} +O(569^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 62 + 115\cdot 569 + 250\cdot 569^{2} + 211\cdot 569^{3} + 103\cdot 569^{4} +O(569^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 79 + 100\cdot 569 + 53\cdot 569^{2} + 318\cdot 569^{3} + 147\cdot 569^{4} +O(569^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 214 + 270\cdot 569 + 152\cdot 569^{2} + 40\cdot 569^{3} + 20\cdot 569^{4} +O(569^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 286 + 427\cdot 569 + 280\cdot 569^{2} + 264\cdot 569^{3} + 480\cdot 569^{4} +O(569^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 474 + 455\cdot 569 + 6\cdot 569^{2} + 65\cdot 569^{3} + 494\cdot 569^{4} +O(569^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 561 + 339\cdot 569 + 82\cdot 569^{2} + 515\cdot 569^{3} + 489\cdot 569^{4} +O(569^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-4$ |
| $2$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(4,5)$ | $0$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,4,3,5)(2,8,7,6)$ | $0$ |
| $2$ | $4$ | $(1,8,3,6)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,7,3,2)(4,8,5,6)$ | $0$ |
| $2$ | $4$ | $(1,8,3,6)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,7)(4,8,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,3,4)(2,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.