Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:C_2^2$ |
| Conductor: | \(1299600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.187142400.1 |
| 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 16.0.2852586422067225600000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 7x^{6} - 18x^{5} + 34x^{4} - 64x^{3} + 78x^{2} - 84x + 61 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8\cdot 61 + 15\cdot 61^{2} + 39\cdot 61^{3} + 46\cdot 61^{4} + 22\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 8\cdot 61 + 19\cdot 61^{2} + 21\cdot 61^{3} + 38\cdot 61^{4} + 28\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 15\cdot 61 + 38\cdot 61^{2} + 47\cdot 61^{3} + 48\cdot 61^{4} + 22\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 + 23\cdot 61 + 29\cdot 61^{2} + 7\cdot 61^{3} + 57\cdot 61^{4} + 50\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 + 26\cdot 61 + 55\cdot 61^{2} + 38\cdot 61^{3} + 8\cdot 61^{4} + 11\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 43 + 18\cdot 61 + 32\cdot 61^{2} + 22\cdot 61^{3} + 28\cdot 61^{4} + 59\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 47 + 16\cdot 61 + 29\cdot 61^{2} + 19\cdot 61^{3} + 61^{4} + 17\cdot 61^{5} +O(61^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 51 + 5\cdot 61 + 25\cdot 61^{2} + 47\cdot 61^{3} + 14\cdot 61^{4} + 31\cdot 61^{5} +O(61^{6})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,2)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(3,4)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,7,2,8)(3,6,4,5)$ | $0$ |
| $2$ | $4$ | $(1,8,2,7)(3,6,4,5)$ | $0$ |
| $2$ | $4$ | $(1,6,2,5)(3,7,4,8)$ | $0$ |
| $2$ | $4$ | $(1,6,2,5)(3,8,4,7)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,7,6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.