Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(306509219201024\)\(\medspace = 2^{15} \cdot 7^{6} \cdot 43^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1568381796352.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T272 |
| Parity: | even |
| Determinant: | 1.344.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.1568381796352.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{6} - 20x^{5} - 72x^{4} - 92x^{3} - 150x^{2} - 152x - 85 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{3} + 4x + 64 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a^{2} + 58 + \left(14 a^{2} + 45 a + 29\right)\cdot 71 + \left(56 a^{2} + 59 a + 42\right)\cdot 71^{2} + \left(39 a^{2} + 59 a + 64\right)\cdot 71^{3} + \left(49 a^{2} + 57 a + 63\right)\cdot 71^{4} + \left(54 a^{2} + 4 a + 67\right)\cdot 71^{5} + \left(70 a^{2} + 19 a + 64\right)\cdot 71^{6} + \left(69 a^{2} + 4 a + 13\right)\cdot 71^{7} + \left(56 a^{2} + 36 a + 11\right)\cdot 71^{8} + \left(33 a^{2} + 58 a + 60\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 58 + 25\cdot 71 + 38\cdot 71^{2} + 53\cdot 71^{3} + 62\cdot 71^{4} + 20\cdot 71^{5} + 16\cdot 71^{6} + 21\cdot 71^{7} + 67\cdot 71^{8} + 18\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 53 + 39\cdot 71 + 66\cdot 71^{2} + 8\cdot 71^{3} + 47\cdot 71^{4} + 5\cdot 71^{5} + 11\cdot 71^{6} + 27\cdot 71^{7} + 16\cdot 71^{8} + 65\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a^{2} + 18 a + 42 + \left(14 a^{2} + 10 a + 49\right)\cdot 71 + \left(10 a^{2} + 62 a + 28\right)\cdot 71^{2} + \left(38 a^{2} + 22 a + 51\right)\cdot 71^{3} + \left(48 a^{2} + 55 a + 42\right)\cdot 71^{4} + \left(15 a^{2} + 27 a + 63\right)\cdot 71^{5} + \left(20 a^{2} + 28 a + 2\right)\cdot 71^{6} + \left(14 a^{2} + 16 a + 29\right)\cdot 71^{7} + \left(17 a^{2} + 63 a + 40\right)\cdot 71^{8} + \left(46 a^{2} + 6 a + 30\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a^{2} + 52 a + 9 + \left(30 a^{2} + 47 a + 67\right)\cdot 71 + \left(35 a^{2} + 38 a + 24\right)\cdot 71^{2} + \left(32 a^{2} + 63 a + 36\right)\cdot 71^{3} + \left(65 a^{2} + 3 a + 40\right)\cdot 71^{4} + \left(30 a^{2} + 5 a + 9\right)\cdot 71^{5} + \left(46 a^{2} + 42 a + 49\right)\cdot 71^{6} + \left(70 a^{2} + 29 a + 13\right)\cdot 71^{7} + \left(3 a^{2} + 8 a + 5\right)\cdot 71^{8} + \left(51 a^{2} + 45 a + 67\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a^{2} + 23 a + 52 + \left(53 a^{2} + 3 a + 14\right)\cdot 71 + \left(27 a^{2} + 7 a + 61\right)\cdot 71^{2} + \left(51 a^{2} + 16 a\right)\cdot 71^{3} + \left(43 a^{2} + 68 a + 1\right)\cdot 71^{4} + \left(14 a^{2} + 15 a + 32\right)\cdot 71^{5} + \left(48 a^{2} + 44 a + 28\right)\cdot 71^{6} + \left(42 a^{2} + 49 a + 59\right)\cdot 71^{7} + \left(29 a^{2} + 6 a + 32\right)\cdot 71^{8} + \left(68 a^{2} + 3 a + 10\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a^{2} + a + 38 + \left(26 a^{2} + 13 a + 56\right)\cdot 71 + \left(25 a^{2} + 41 a + 21\right)\cdot 71^{2} + \left(55 a + 45\right)\cdot 71^{3} + \left(28 a^{2} + 11 a + 11\right)\cdot 71^{4} + \left(24 a^{2} + 38 a + 63\right)\cdot 71^{5} + \left(4 a^{2} + 7\right)\cdot 71^{6} + \left(57 a^{2} + 25 a + 1\right)\cdot 71^{7} + \left(49 a^{2} + 70 a + 9\right)\cdot 71^{8} + \left(44 a^{2} + 18 a + 50\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 33 a^{2} + 48 a + 45 + \left(3 a^{2} + 22 a\right)\cdot 71 + \left(58 a^{2} + 4 a\right)\cdot 71^{2} + \left(50 a^{2} + 66 a + 23\right)\cdot 71^{3} + \left(48 a^{2} + 15 a + 14\right)\cdot 71^{4} + \left(a^{2} + 50 a + 21\right)\cdot 71^{5} + \left(23 a^{2} + 7 a + 32\right)\cdot 71^{6} + \left(29 a^{2} + 17 a + 47\right)\cdot 71^{7} + \left(55 a^{2} + 28 a + 30\right)\cdot 71^{8} + \left(39 a^{2} + 9 a + 52\right)\cdot 71^{9} +O(71^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,6)(2,8)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $1$ |
| $12$ | $2$ | $(3,4)$ | $-3$ |
| $24$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $36$ | $2$ | $(1,6)(2,8)(3,4)$ | $1$ |
| $16$ | $3$ | $(3,5,7)$ | $0$ |
| $64$ | $3$ | $(2,6,8)(3,5,7)$ | $0$ |
| $12$ | $4$ | $(1,2,6,8)$ | $3$ |
| $36$ | $4$ | $(1,2,6,8)(3,4,5,7)$ | $1$ |
| $36$ | $4$ | $(1,6)(2,8)(3,4,5,7)$ | $-1$ |
| $72$ | $4$ | $(1,5,6,3)(2,7,8,4)$ | $-1$ |
| $72$ | $4$ | $(1,2,6,8)(3,4)$ | $-1$ |
| $144$ | $4$ | $(1,3,2,4)(5,6)(7,8)$ | $-1$ |
| $48$ | $6$ | $(1,6)(2,8)(3,7,5)$ | $0$ |
| $96$ | $6$ | $(2,8,6)(3,4)$ | $0$ |
| $192$ | $6$ | $(1,4)(2,5,6,7,8,3)$ | $0$ |
| $144$ | $8$ | $(1,4,2,5,6,7,8,3)$ | $1$ |
| $96$ | $12$ | $(1,2,6,8)(3,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.