Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(262\!\cdots\!000\)\(\medspace = 2^{51} \cdot 3^{14} \cdot 5^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.119439360000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.119439360000.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{6} - 12x^{5} + 6x^{4} + 48x^{3} - 32x^{2} + 16 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$:
\( x^{3} + 3x + 86 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 43 a^{2} + 8 a + 22 + \left(17 a^{2} + 48 a + 34\right)\cdot 89 + \left(23 a^{2} + 49 a + 71\right)\cdot 89^{2} + \left(14 a^{2} + 71 a + 5\right)\cdot 89^{3} + \left(47 a^{2} + 43 a + 54\right)\cdot 89^{4} + \left(36 a^{2} + 32 a + 73\right)\cdot 89^{5} + \left(61 a^{2} + 53 a + 27\right)\cdot 89^{6} + \left(48 a^{2} + 34 a + 65\right)\cdot 89^{7} + \left(59 a + 23\right)\cdot 89^{8} + \left(63 a^{2} + 51 a + 77\right)\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 5\cdot 89 + 60\cdot 89^{2} + 41\cdot 89^{3} + 8\cdot 89^{4} + 86\cdot 89^{5} + 43\cdot 89^{6} + 18\cdot 89^{7} + 8\cdot 89^{8} + 88\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a^{2} + 73 a + 73 + \left(67 a^{2} + 36 a + 44\right)\cdot 89 + \left(75 a^{2} + 62 a + 87\right)\cdot 89^{2} + \left(88 a^{2} + 27 a + 65\right)\cdot 89^{3} + \left(80 a^{2} + 47 a + 32\right)\cdot 89^{4} + \left(79 a^{2} + 5 a + 71\right)\cdot 89^{5} + \left(68 a^{2} + 60 a + 42\right)\cdot 89^{6} + \left(69 a^{2} + 18\right)\cdot 89^{7} + \left(61 a^{2} + 71 a + 57\right)\cdot 89^{8} + \left(27 a^{2} + 32 a + 6\right)\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 78 + 67\cdot 89 + 63\cdot 89^{2} + 18\cdot 89^{3} + 6\cdot 89^{4} + 79\cdot 89^{5} + 77\cdot 89^{6} + 3\cdot 89^{7} + 11\cdot 89^{8} + 21\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a^{2} + 8 a + 69 + \left(4 a^{2} + 4 a + 7\right)\cdot 89 + \left(79 a^{2} + 66 a + 5\right)\cdot 89^{2} + \left(74 a^{2} + 78 a + 38\right)\cdot 89^{3} + \left(49 a^{2} + 86 a + 59\right)\cdot 89^{4} + \left(61 a^{2} + 50 a + 34\right)\cdot 89^{5} + \left(47 a^{2} + 64 a\right)\cdot 89^{6} + \left(59 a^{2} + 53 a + 87\right)\cdot 89^{7} + \left(26 a^{2} + 47 a + 75\right)\cdot 89^{8} + \left(87 a^{2} + 4 a + 36\right)\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 a^{2} + 57 a + 14 + \left(78 a^{2} + 16 a + 14\right)\cdot 89 + \left(35 a^{2} + 41 a + 35\right)\cdot 89^{2} + \left(9 a^{2} + 16 a + 21\right)\cdot 89^{3} + \left(35 a^{2} + 40 a + 46\right)\cdot 89^{4} + \left(22 a^{2} + 60 a + 48\right)\cdot 89^{5} + \left(3 a^{2} + 86 a + 1\right)\cdot 89^{6} + \left(55 a^{2} + 2 a + 16\right)\cdot 89^{7} + \left(8 a^{2} + 77 a + 77\right)\cdot 89^{8} + \left(41 a^{2} + 87 a + 64\right)\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 25 a^{2} + 36 a + 22 + \left(56 a^{2} + 77 a + 59\right)\cdot 89 + \left(73 a^{2} + 48 a + 21\right)\cdot 89^{2} + \left(21 a^{2} + 85 a + 46\right)\cdot 89^{3} + \left(71 a^{2} + 13 a + 29\right)\cdot 89^{4} + \left(67 a^{2} + 78 a + 50\right)\cdot 89^{5} + \left(83 a^{2} + 56 a + 73\right)\cdot 89^{6} + \left(71 a^{2} + 61 a + 49\right)\cdot 89^{7} + \left(16 a^{2} + 60 a + 4\right)\cdot 89^{8} + \left(16 a^{2} + 7 a + 15\right)\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 43 a^{2} + 85 a + 58 + \left(43 a^{2} + 83 a + 33\right)\cdot 89 + \left(68 a^{2} + 87 a + 11\right)\cdot 89^{2} + \left(57 a^{2} + 75 a + 29\right)\cdot 89^{3} + \left(71 a^{2} + 34 a + 30\right)\cdot 89^{4} + \left(87 a^{2} + 39 a + 1\right)\cdot 89^{5} + \left(a^{2} + 34 a + 88\right)\cdot 89^{6} + \left(51 a^{2} + 24 a + 7\right)\cdot 89^{7} + \left(63 a^{2} + 40 a + 9\right)\cdot 89^{8} + \left(31 a^{2} + 82 a + 46\right)\cdot 89^{9} +O(89^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,7)(6,8)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,6)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,7)(6,8)$ | $0$ |
| $16$ | $3$ | $(1,4,5)$ | $0$ |
| $64$ | $3$ | $(1,4,5)(6,7,8)$ | $0$ |
| $12$ | $4$ | $(2,6,7,8)$ | $0$ |
| $36$ | $4$ | $(1,3,4,5)(2,6,7,8)$ | $-2$ |
| $36$ | $4$ | $(1,3,4,5)(2,7)(6,8)$ | $0$ |
| $72$ | $4$ | $(1,2,4,7)(3,6,5,8)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,6,7,8)$ | $2$ |
| $144$ | $4$ | $(1,6,3,2)(4,7)(5,8)$ | $0$ |
| $48$ | $6$ | $(1,5,4)(2,7)(6,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(6,8,7)$ | $0$ |
| $192$ | $6$ | $(1,6,4,7,5,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,6,4,7,5,8)$ | $0$ |
| $96$ | $12$ | $(1,4,5)(2,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.