Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(417\!\cdots\!696\)\(\medspace = 2^{39} \cdot 97^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.239251750912.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | even |
| Determinant: | 1.776.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.239251750912.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} - 4x^{5} + 10x^{4} - 12x^{3} - 10x^{2} + 16x + 7 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 21 a + 43 + \left(19 a + 23\right)\cdot 71 + \left(5 a + 33\right)\cdot 71^{2} + \left(30 a + 51\right)\cdot 71^{3} + \left(70 a + 19\right)\cdot 71^{4} + \left(61 a + 5\right)\cdot 71^{5} + \left(63 a + 2\right)\cdot 71^{6} + \left(42 a + 60\right)\cdot 71^{7} + \left(55 a + 25\right)\cdot 71^{8} + \left(42 a + 37\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 62 + 53\cdot 71 + 70\cdot 71^{2} + 43\cdot 71^{3} + 36\cdot 71^{4} + 7\cdot 71^{5} + 26\cdot 71^{6} + 44\cdot 71^{7} + 8\cdot 71^{8} + 49\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 a + 31 + \left(13 a + 48\right)\cdot 71 + \left(a + 27\right)\cdot 71^{2} + \left(16 a + 58\right)\cdot 71^{3} + \left(69 a + 11\right)\cdot 71^{4} + \left(36 a + 24\right)\cdot 71^{5} + \left(27 a + 13\right)\cdot 71^{6} + \left(52 a + 69\right)\cdot 71^{7} + \left(56 a + 3\right)\cdot 71^{8} + 19 a\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 a + 14 + \left(51 a + 41\right)\cdot 71 + \left(65 a + 24\right)\cdot 71^{2} + \left(40 a + 35\right)\cdot 71^{3} + 59\cdot 71^{4} + \left(9 a + 58\right)\cdot 71^{5} + \left(7 a + 67\right)\cdot 71^{6} + \left(28 a + 10\right)\cdot 71^{7} + \left(15 a + 23\right)\cdot 71^{8} + \left(28 a + 67\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 67 + \left(56 a + 66\right)\cdot 71 + \left(15 a + 46\right)\cdot 71^{2} + \left(22 a + 61\right)\cdot 71^{3} + \left(48 a + 36\right)\cdot 71^{4} + \left(6 a + 10\right)\cdot 71^{5} + \left(66 a + 32\right)\cdot 71^{6} + \left(35 a + 48\right)\cdot 71^{7} + \left(10 a + 39\right)\cdot 71^{8} + 13\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 66 + 65\cdot 71 + 41\cdot 71^{2} + 66\cdot 71^{3} + 15\cdot 71^{4} + 31\cdot 71^{5} + 24\cdot 71^{6} + 63\cdot 71^{7} + 21\cdot 71^{8} + 59\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 54 a + 30 + \left(14 a + 20\right)\cdot 71 + \left(55 a + 22\right)\cdot 71^{2} + \left(48 a + 19\right)\cdot 71^{3} + \left(22 a + 40\right)\cdot 71^{4} + \left(64 a + 46\right)\cdot 71^{5} + \left(4 a + 15\right)\cdot 71^{6} + \left(35 a + 54\right)\cdot 71^{7} + \left(60 a + 24\right)\cdot 71^{8} + \left(70 a + 3\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 30 a + 42 + \left(57 a + 34\right)\cdot 71 + \left(69 a + 16\right)\cdot 71^{2} + \left(54 a + 18\right)\cdot 71^{3} + \left(a + 63\right)\cdot 71^{4} + \left(34 a + 28\right)\cdot 71^{5} + \left(43 a + 31\right)\cdot 71^{6} + \left(18 a + 4\right)\cdot 71^{7} + \left(14 a + 65\right)\cdot 71^{8} + \left(51 a + 53\right)\cdot 71^{9} +O(71^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,4)(3,8)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,6)(5,7)$ | $0$ |
| $16$ | $3$ | $(3,4,8)$ | $0$ |
| $64$ | $3$ | $(3,4,8)(5,6,7)$ | $0$ |
| $12$ | $4$ | $(1,3,4,8)$ | $0$ |
| $36$ | $4$ | $(1,3,4,8)(2,5,6,7)$ | $-2$ |
| $36$ | $4$ | $(1,4)(2,5,6,7)(3,8)$ | $0$ |
| $72$ | $4$ | $(1,6,4,2)(3,7,8,5)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,5,6,7)$ | $2$ |
| $144$ | $4$ | $(1,5,3,2)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(2,6)(3,8,4)(5,7)$ | $0$ |
| $96$ | $6$ | $(2,5)(3,4,8)$ | $0$ |
| $192$ | $6$ | $(1,2)(3,6,4,7,8,5)$ | $0$ |
| $144$ | $8$ | $(1,5,3,6,4,7,8,2)$ | $0$ |
| $96$ | $12$ | $(2,5,6,7)(3,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.