Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(136161\)\(\medspace = 3^{4} \cdot 41^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.150730227.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8:S_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2:S_4$ |
| Projective stem field: | Galois closure of 8.0.31165433925201.4 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 2x^{6} + 3x^{5} + 6x^{4} - 3x^{3} - 10x^{2} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 41\cdot 67 + 34\cdot 67^{2} + 41\cdot 67^{3} + 16\cdot 67^{4} + 21\cdot 67^{5} + 15\cdot 67^{6} + 54\cdot 67^{7} + 49\cdot 67^{8} + 56\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 25 + \left(38 a + 15\right)\cdot 67 + \left(60 a + 22\right)\cdot 67^{2} + \left(42 a + 13\right)\cdot 67^{3} + \left(43 a + 25\right)\cdot 67^{4} + \left(63 a + 59\right)\cdot 67^{5} + \left(30 a + 65\right)\cdot 67^{6} + \left(48 a + 54\right)\cdot 67^{7} + \left(13 a + 16\right)\cdot 67^{8} + \left(29 a + 54\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 + 12\cdot 67^{2} + 56\cdot 67^{3} + 23\cdot 67^{4} + 55\cdot 67^{5} + 30\cdot 67^{6} + 15\cdot 67^{7} + 36\cdot 67^{8} + 36\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 a + 2 + \left(3 a + 51\right)\cdot 67 + \left(65 a + 56\right)\cdot 67^{2} + \left(24 a + 42\right)\cdot 67^{3} + \left(46 a + 6\right)\cdot 67^{4} + \left(60 a + 52\right)\cdot 67^{5} + \left(10 a + 53\right)\cdot 67^{6} + \left(61 a + 24\right)\cdot 67^{7} + \left(55 a + 14\right)\cdot 67^{8} + \left(31 a + 41\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 a + 38 + \left(28 a + 14\right)\cdot 67 + \left(6 a + 25\right)\cdot 67^{2} + \left(24 a + 57\right)\cdot 67^{3} + \left(23 a + 22\right)\cdot 67^{4} + \left(3 a + 2\right)\cdot 67^{5} + \left(36 a + 59\right)\cdot 67^{6} + \left(18 a + 16\right)\cdot 67^{7} + \left(53 a + 23\right)\cdot 67^{8} + \left(37 a + 23\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 40 + \left(63 a + 22\right)\cdot 67 + \left(a + 45\right)\cdot 67^{2} + \left(42 a + 10\right)\cdot 67^{3} + \left(20 a + 33\right)\cdot 67^{4} + \left(6 a + 47\right)\cdot 67^{5} + \left(56 a + 36\right)\cdot 67^{6} + \left(5 a + 57\right)\cdot 67^{7} + \left(11 a + 42\right)\cdot 67^{8} + \left(35 a + 45\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 62 a + \left(48 a + 28\right)\cdot 67 + \left(15 a + 62\right)\cdot 67^{2} + \left(10 a + 43\right)\cdot 67^{3} + \left(32 a + 10\right)\cdot 67^{4} + \left(19 a + 59\right)\cdot 67^{5} + \left(10 a + 25\right)\cdot 67^{6} + \left(2 a + 56\right)\cdot 67^{7} + \left(30 a + 16\right)\cdot 67^{8} + \left(31 a + 24\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 5 a + 47 + \left(18 a + 27\right)\cdot 67 + \left(51 a + 9\right)\cdot 67^{2} + \left(56 a + 2\right)\cdot 67^{3} + \left(34 a + 62\right)\cdot 67^{4} + \left(47 a + 37\right)\cdot 67^{5} + \left(56 a + 47\right)\cdot 67^{6} + \left(64 a + 54\right)\cdot 67^{7} + 36 a\cdot 67^{8} + \left(35 a + 53\right)\cdot 67^{9} +O(67^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $-4$ |
| $6$ | $2$ | $(1,3)(2,6)$ | $0$ |
| $12$ | $2$ | $(1,6)(2,3)(4,7)(5,8)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,6)(7,8)$ | $0$ |
| $32$ | $3$ | $(1,5,7)(3,4,8)$ | $1$ |
| $6$ | $4$ | $(1,5,3,4)(2,8,6,7)$ | $0$ |
| $6$ | $4$ | $(1,8,3,7)(2,5,6,4)$ | $0$ |
| $12$ | $4$ | $(1,2,3,6)(4,5)(7,8)$ | $-2$ |
| $12$ | $4$ | $(2,4,6,5)$ | $2$ |
| $32$ | $6$ | $(1,8,5,3,7,4)(2,6)$ | $-1$ |
| $24$ | $8$ | $(1,2,8,5,3,6,7,4)$ | $0$ |
| $24$ | $8$ | $(1,2,8,4,3,6,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.