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 8.0.768144384.2 |
| 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 8.4.197605142784.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 2x^{6} + 2x^{5} + 8x^{4} + 14x^{3} + 14x^{2} + 8x + 2 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 25 a + 20 + \left(11 a + 30\right)\cdot 71 + \left(67 a + 24\right)\cdot 71^{2} + \left(59 a + 53\right)\cdot 71^{3} + \left(8 a + 45\right)\cdot 71^{4} + \left(4 a + 35\right)\cdot 71^{5} + \left(40 a + 56\right)\cdot 71^{6} + \left(48 a + 36\right)\cdot 71^{7} + \left(27 a + 61\right)\cdot 71^{8} + \left(29 a + 20\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 43 a + 70 + \left(6 a + 19\right)\cdot 71 + \left(21 a + 9\right)\cdot 71^{2} + \left(56 a + 53\right)\cdot 71^{3} + \left(5 a + 55\right)\cdot 71^{4} + \left(31 a + 28\right)\cdot 71^{5} + \left(35 a + 51\right)\cdot 71^{6} + \left(2 a + 56\right)\cdot 71^{7} + \left(13 a + 35\right)\cdot 71^{8} + \left(59 a + 5\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 52 a + 12 + \left(45 a + 38\right)\cdot 71 + \left(7 a + 24\right)\cdot 71^{2} + \left(44 a + 36\right)\cdot 71^{3} + \left(20 a + 54\right)\cdot 71^{4} + \left(63 a + 68\right)\cdot 71^{5} + \left(59 a + 4\right)\cdot 71^{6} + \left(16 a + 16\right)\cdot 71^{7} + \left(64 a + 60\right)\cdot 71^{8} + \left(51 a + 38\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 5\cdot 71 + 62\cdot 71^{2} + 37\cdot 71^{3} + 15\cdot 71^{4} + 14\cdot 71^{5} + 30\cdot 71^{6} + 58\cdot 71^{7} + 46\cdot 71^{8} + 8\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 a + 70 + \left(59 a + 27\right)\cdot 71 + \left(3 a + 5\right)\cdot 71^{2} + \left(11 a + 35\right)\cdot 71^{3} + \left(62 a + 3\right)\cdot 71^{4} + \left(66 a + 35\right)\cdot 71^{5} + \left(30 a + 61\right)\cdot 71^{6} + \left(22 a + 22\right)\cdot 71^{7} + \left(43 a + 68\right)\cdot 71^{8} + \left(41 a + 51\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 + 23\cdot 71 + 48\cdot 71^{2} + 19\cdot 71^{3} + 46\cdot 71^{4} + 54\cdot 71^{5} + 68\cdot 71^{6} + 5\cdot 71^{7} + 64\cdot 71^{8} + 39\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 a + 14 + \left(64 a + 61\right)\cdot 71 + \left(49 a + 44\right)\cdot 71^{2} + \left(14 a + 2\right)\cdot 71^{3} + \left(65 a + 11\right)\cdot 71^{4} + \left(39 a + 14\right)\cdot 71^{5} + \left(35 a + 20\right)\cdot 71^{6} + \left(68 a + 26\right)\cdot 71^{7} + \left(57 a + 59\right)\cdot 71^{8} + \left(11 a + 39\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 19 a + 45 + \left(25 a + 6\right)\cdot 71 + \left(63 a + 65\right)\cdot 71^{2} + \left(26 a + 45\right)\cdot 71^{3} + \left(50 a + 51\right)\cdot 71^{4} + \left(7 a + 32\right)\cdot 71^{5} + \left(11 a + 61\right)\cdot 71^{6} + \left(54 a + 60\right)\cdot 71^{7} + \left(6 a + 29\right)\cdot 71^{8} + \left(19 a + 7\right)\cdot 71^{9} +O(71^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $-4$ |
| $6$ | $2$ | $(3,8)(4,6)$ | $0$ |
| $12$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(7,8)$ | $0$ |
| $32$ | $3$ | $(1,6,3)(2,4,8)$ | $1$ |
| $6$ | $4$ | $(1,8,2,3)(4,7,6,5)$ | $0$ |
| $6$ | $4$ | $(1,4,2,6)(3,5,8,7)$ | $0$ |
| $12$ | $4$ | $(1,5,2,7)$ | $2$ |
| $12$ | $4$ | $(1,5,2,7)(3,8)(4,6)$ | $-2$ |
| $32$ | $6$ | $(1,8,6,2,3,4)(5,7)$ | $-1$ |
| $24$ | $8$ | $(1,7,8,6,2,5,3,4)$ | $0$ |
| $24$ | $8$ | $(1,7,4,3,2,5,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.