Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(219024\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.922529088.1 |
| 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 8.4.8107185625344.5 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 8x^{6} - 2x^{5} + 8x^{4} + 2x^{3} + 8x^{2} - 2x + 4 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 20 a + 60 + \left(23 a + 33\right)\cdot 67 + \left(59 a + 60\right)\cdot 67^{2} + \left(50 a + 43\right)\cdot 67^{3} + 45 a\cdot 67^{4} + \left(4 a + 9\right)\cdot 67^{5} + \left(32 a + 58\right)\cdot 67^{6} + \left(26 a + 23\right)\cdot 67^{7} + \left(35 a + 13\right)\cdot 67^{8} + \left(54 a + 5\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 52 a + 30 + \left(4 a + 23\right)\cdot 67 + \left(50 a + 34\right)\cdot 67^{2} + \left(16 a + 54\right)\cdot 67^{3} + \left(5 a + 6\right)\cdot 67^{4} + \left(63 a + 32\right)\cdot 67^{5} + \left(18 a + 4\right)\cdot 67^{6} + \left(63 a + 8\right)\cdot 67^{7} + \left(52 a + 26\right)\cdot 67^{8} + \left(39 a + 7\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 + 62\cdot 67 + 50\cdot 67^{2} + 30\cdot 67^{3} + 4\cdot 67^{4} + 47\cdot 67^{5} + 19\cdot 67^{6} + 36\cdot 67^{7} + 12\cdot 67^{8} + 15\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 a + 6 + \left(43 a + 40\right)\cdot 67 + \left(7 a + 6\right)\cdot 67^{2} + \left(16 a + 54\right)\cdot 67^{3} + \left(21 a + 65\right)\cdot 67^{4} + \left(62 a + 48\right)\cdot 67^{5} + \left(34 a + 47\right)\cdot 67^{6} + \left(40 a + 30\right)\cdot 67^{7} + \left(31 a + 61\right)\cdot 67^{8} + \left(12 a + 53\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 33 + \left(4 a + 3\right)\cdot 67 + \left(61 a + 50\right)\cdot 67^{2} + \left(52 a + 55\right)\cdot 67^{3} + \left(52 a + 37\right)\cdot 67^{4} + \left(7 a + 35\right)\cdot 67^{5} + \left(39 a + 3\right)\cdot 67^{6} + \left(13 a + 36\right)\cdot 67^{7} + \left(65 a + 29\right)\cdot 67^{8} + \left(20 a + 23\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 37 + \left(62 a + 57\right)\cdot 67 + \left(16 a + 28\right)\cdot 67^{2} + \left(50 a + 4\right)\cdot 67^{3} + \left(61 a + 11\right)\cdot 67^{4} + \left(3 a + 11\right)\cdot 67^{5} + \left(48 a + 17\right)\cdot 67^{6} + \left(3 a + 41\right)\cdot 67^{7} + \left(14 a + 40\right)\cdot 67^{8} + \left(27 a + 46\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 64 a + 45 + \left(62 a + 16\right)\cdot 67 + \left(5 a + 22\right)\cdot 67^{2} + \left(14 a + 5\right)\cdot 67^{3} + \left(14 a + 62\right)\cdot 67^{4} + \left(59 a + 13\right)\cdot 67^{5} + \left(27 a + 18\right)\cdot 67^{6} + \left(53 a + 51\right)\cdot 67^{7} + \left(a + 8\right)\cdot 67^{8} + \left(46 a + 42\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 21 + 30\cdot 67 + 14\cdot 67^{2} + 19\cdot 67^{3} + 12\cdot 67^{4} + 3\cdot 67^{5} + 32\cdot 67^{6} + 40\cdot 67^{7} + 8\cdot 67^{8} + 7\cdot 67^{9} +O(67^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $-4$ |
| $6$ | $2$ | $(2,4)(5,7)$ | $0$ |
| $12$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $0$ |
| $24$ | $2$ | $(1,4)(2,6)(3,8)$ | $0$ |
| $32$ | $3$ | $(1,8,7)(3,5,6)$ | $1$ |
| $6$ | $4$ | $(1,3,6,8)(2,7,4,5)$ | $0$ |
| $6$ | $4$ | $(1,5,6,7)(2,8,4,3)$ | $0$ |
| $12$ | $4$ | $(1,4,6,2)(3,8)(5,7)$ | $-2$ |
| $12$ | $4$ | $(2,7,4,5)$ | $2$ |
| $32$ | $6$ | $(1,5,8,6,7,3)(2,4)$ | $-1$ |
| $24$ | $8$ | $(1,4,5,3,6,2,7,8)$ | $0$ |
| $24$ | $8$ | $(1,4,7,3,6,2,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.