Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(128\!\cdots\!759\)\(\medspace = 3^{22} \cdot 151^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.7529733837.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.151.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.7529733837.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} - x^{5} + 24x^{4} + 3x^{3} - 41x^{2} + 27 \)
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The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$:
\( x^{2} + 108x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 62 a + 19 + \left(58 a + 54\right)\cdot 109 + \left(28 a + 104\right)\cdot 109^{2} + \left(30 a + 79\right)\cdot 109^{3} + \left(48 a + 70\right)\cdot 109^{4} + \left(52 a + 95\right)\cdot 109^{5} + \left(68 a + 97\right)\cdot 109^{6} + \left(27 a + 76\right)\cdot 109^{7} + \left(14 a + 56\right)\cdot 109^{8} + \left(89 a + 36\right)\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 8 + \left(41 a + 46\right)\cdot 109 + \left(2 a + 1\right)\cdot 109^{2} + \left(84 a + 106\right)\cdot 109^{3} + \left(15 a + 93\right)\cdot 109^{4} + \left(21 a + 47\right)\cdot 109^{5} + \left(45 a + 49\right)\cdot 109^{6} + \left(82 a + 78\right)\cdot 109^{7} + \left(79 a + 58\right)\cdot 109^{8} + \left(39 a + 1\right)\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 95 + 57\cdot 109 + 59\cdot 109^{2} + 99\cdot 109^{3} + 17\cdot 109^{4} + 65\cdot 109^{5} + 28\cdot 109^{6} + 58\cdot 109^{7} + 83\cdot 109^{8} + 46\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 53 + \left(67 a + 32\right)\cdot 109 + \left(21 a + 97\right)\cdot 109^{2} + \left(59 a + 35\right)\cdot 109^{3} + \left(59 a + 9\right)\cdot 109^{4} + \left(63 a + 29\right)\cdot 109^{5} + \left(11 a + 37\right)\cdot 109^{6} + \left(12 a + 93\right)\cdot 109^{7} + \left(89 a + 21\right)\cdot 109^{8} + \left(98 a + 50\right)\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 a + 81 + \left(50 a + 50\right)\cdot 109 + \left(80 a + 74\right)\cdot 109^{2} + \left(78 a + 81\right)\cdot 109^{3} + \left(60 a + 88\right)\cdot 109^{4} + \left(56 a + 99\right)\cdot 109^{5} + \left(40 a + 4\right)\cdot 109^{6} + \left(81 a + 36\right)\cdot 109^{7} + \left(94 a + 43\right)\cdot 109^{8} + \left(19 a + 2\right)\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 98 a + 19 + \left(67 a + 76\right)\cdot 109 + \left(106 a + 71\right)\cdot 109^{2} + \left(24 a + 78\right)\cdot 109^{3} + \left(93 a + 25\right)\cdot 109^{4} + \left(87 a + 53\right)\cdot 109^{5} + \left(63 a + 73\right)\cdot 109^{6} + \left(26 a + 6\right)\cdot 109^{7} + \left(29 a + 56\right)\cdot 109^{8} + \left(69 a + 70\right)\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 81 + 45\cdot 109 + 84\cdot 109^{2} + 98\cdot 109^{3} + 10\cdot 109^{4} + 12\cdot 109^{5} + 50\cdot 109^{6} + 101\cdot 109^{7} + 16\cdot 109^{8} + 59\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 82 a + 80 + \left(41 a + 72\right)\cdot 109 + \left(87 a + 51\right)\cdot 109^{2} + \left(49 a + 73\right)\cdot 109^{3} + \left(49 a + 9\right)\cdot 109^{4} + \left(45 a + 33\right)\cdot 109^{5} + \left(97 a + 94\right)\cdot 109^{6} + \left(96 a + 93\right)\cdot 109^{7} + \left(19 a + 98\right)\cdot 109^{8} + \left(10 a + 59\right)\cdot 109^{9} +O(109^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $-6$ |
| $9$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $2$ |
| $12$ | $2$ | $(3,4)$ | $0$ |
| $24$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $36$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $36$ | $2$ | $(1,5)(2,6)(3,4)$ | $0$ |
| $16$ | $3$ | $(3,7,8)$ | $0$ |
| $64$ | $3$ | $(2,5,6)(3,7,8)$ | $0$ |
| $12$ | $4$ | $(1,2,5,6)$ | $0$ |
| $36$ | $4$ | $(1,2,5,6)(3,4,7,8)$ | $-2$ |
| $36$ | $4$ | $(1,5)(2,6)(3,4,7,8)$ | $0$ |
| $72$ | $4$ | $(1,7,5,3)(2,8,6,4)$ | $0$ |
| $72$ | $4$ | $(1,2,5,6)(3,4)$ | $2$ |
| $144$ | $4$ | $(1,3,2,4)(5,7)(6,8)$ | $0$ |
| $48$ | $6$ | $(1,5)(2,6)(3,8,7)$ | $0$ |
| $96$ | $6$ | $(2,6,5)(3,4)$ | $0$ |
| $192$ | $6$ | $(1,4)(2,7,5,8,6,3)$ | $0$ |
| $144$ | $8$ | $(1,4,2,7,5,8,6,3)$ | $0$ |
| $96$ | $12$ | $(1,2,5,6)(3,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.