Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(254016\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.768144384.1 |
| 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 8.4.197605142784.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} - 12x^{5} + 6x^{4} - 6x^{3} - 6x^{2} + 12x + 6 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 51 + 71 + 33\cdot 71^{2} + 48\cdot 71^{3} + 71^{4} + 7\cdot 71^{5} + 40\cdot 71^{6} + 58\cdot 71^{7} + 51\cdot 71^{8} + 18\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 10 + \left(37 a + 10\right)\cdot 71 + \left(4 a + 68\right)\cdot 71^{2} + \left(52 a + 37\right)\cdot 71^{3} + \left(14 a + 8\right)\cdot 71^{4} + \left(31 a + 23\right)\cdot 71^{5} + \left(32 a + 40\right)\cdot 71^{6} + \left(3 a + 27\right)\cdot 71^{7} + \left(8 a + 16\right)\cdot 71^{8} + \left(41 a + 37\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 + 56\cdot 71 + 13\cdot 71^{2} + 12\cdot 71^{3} + 4\cdot 71^{4} + 61\cdot 71^{5} + 5\cdot 71^{6} + 23\cdot 71^{7} + 64\cdot 71^{8} + 70\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a + 46 + \left(68 a + 61\right)\cdot 71 + \left(64 a + 36\right)\cdot 71^{2} + \left(6 a + 30\right)\cdot 71^{3} + \left(64 a + 47\right)\cdot 71^{4} + \left(18 a + 43\right)\cdot 71^{5} + \left(31 a + 46\right)\cdot 71^{6} + \left(2 a + 9\right)\cdot 71^{7} + \left(65 a + 7\right)\cdot 71^{8} + \left(20 a + 3\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 66 a + 20 + \left(33 a + 8\right)\cdot 71 + \left(66 a + 40\right)\cdot 71^{2} + \left(18 a + 66\right)\cdot 71^{3} + \left(56 a + 56\right)\cdot 71^{4} + \left(39 a + 70\right)\cdot 71^{5} + \left(38 a + 2\right)\cdot 71^{6} + \left(67 a + 2\right)\cdot 71^{7} + \left(62 a + 29\right)\cdot 71^{8} + \left(29 a + 40\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 41 a + 35 + \left(2 a + 26\right)\cdot 71 + \left(6 a + 27\right)\cdot 71^{2} + \left(64 a + 50\right)\cdot 71^{3} + \left(6 a + 26\right)\cdot 71^{4} + \left(52 a + 17\right)\cdot 71^{5} + \left(39 a + 19\right)\cdot 71^{6} + \left(68 a + 54\right)\cdot 71^{7} + \left(5 a + 63\right)\cdot 71^{8} + \left(50 a + 50\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 55 a + 56 + \left(42 a + 8\right)\cdot 71 + \left(19 a + 34\right)\cdot 71^{2} + \left(60 a + 39\right)\cdot 71^{3} + \left(19 a + 8\right)\cdot 71^{4} + 40 a\cdot 71^{5} + \left(46 a + 38\right)\cdot 71^{6} + \left(23 a + 18\right)\cdot 71^{7} + \left(35 a + 2\right)\cdot 71^{8} + \left(36 a + 48\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 16 a + 24 + \left(28 a + 39\right)\cdot 71 + \left(51 a + 30\right)\cdot 71^{2} + \left(10 a + 69\right)\cdot 71^{3} + \left(51 a + 58\right)\cdot 71^{4} + \left(30 a + 60\right)\cdot 71^{5} + \left(24 a + 19\right)\cdot 71^{6} + \left(47 a + 19\right)\cdot 71^{7} + \left(35 a + 49\right)\cdot 71^{8} + \left(34 a + 14\right)\cdot 71^{9} +O(71^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $-4$ |
| $6$ | $2$ | $(1,3)(7,8)$ | $0$ |
| $12$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $0$ |
| $24$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $32$ | $3$ | $(1,6,7)(3,5,8)$ | $1$ |
| $6$ | $4$ | $(1,2,3,4)(5,8,6,7)$ | $0$ |
| $6$ | $4$ | $(1,5,3,6)(2,8,4,7)$ | $0$ |
| $12$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $-2$ |
| $12$ | $4$ | $(2,8,4,7)$ | $2$ |
| $32$ | $6$ | $(1,5,2,3,6,4)(7,8)$ | $-1$ |
| $24$ | $8$ | $(1,4,5,7,3,2,6,8)$ | $0$ |
| $24$ | $8$ | $(1,8,4,5,3,7,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.