Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(417\!\cdots\!568\)\(\medspace = 2^{51} \cdot 3^{32} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1719926784.1 |
| 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 8.2.1719926784.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 8x^{6} - 8x^{5} - x^{4} + 20x^{3} - 28x^{2} + 16x - 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$:
\( x^{2} + 102x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 75 a + 18 + \left(22 a + 74\right)\cdot 103 + \left(55 a + 7\right)\cdot 103^{2} + \left(58 a + 50\right)\cdot 103^{3} + \left(84 a + 61\right)\cdot 103^{4} + \left(94 a + 28\right)\cdot 103^{5} + \left(84 a + 1\right)\cdot 103^{6} + \left(18 a + 43\right)\cdot 103^{7} + \left(15 a + 73\right)\cdot 103^{8} + \left(3 a + 89\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 83 + 50\cdot 103 + 60\cdot 103^{2} + 67\cdot 103^{3} + 2\cdot 103^{4} + 76\cdot 103^{5} + 44\cdot 103^{6} + 34\cdot 103^{7} + 74\cdot 103^{8} + 38\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 a + 1 + \left(95 a + 1\right)\cdot 103 + \left(22 a + 2\right)\cdot 103^{2} + \left(48 a + 17\right)\cdot 103^{3} + \left(67 a + 26\right)\cdot 103^{4} + \left(16 a + 12\right)\cdot 103^{5} + \left(16 a + 55\right)\cdot 103^{6} + \left(82 a + 20\right)\cdot 103^{7} + \left(47 a + 102\right)\cdot 103^{8} + \left(46 a + 40\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 24 + \left(35 a + 6\right)\cdot 103 + \left(30 a + 29\right)\cdot 103^{2} + \left(13 a + 52\right)\cdot 103^{3} + \left(91 a + 55\right)\cdot 103^{4} + \left(43 a + 54\right)\cdot 103^{5} + \left(70 a + 3\right)\cdot 103^{6} + \left(29 a + 76\right)\cdot 103^{7} + \left(84 a + 86\right)\cdot 103^{8} + \left(44 a + 98\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 92 a + 35 + \left(67 a + 30\right)\cdot 103 + \left(72 a + 24\right)\cdot 103^{2} + \left(89 a + 35\right)\cdot 103^{3} + \left(11 a + 30\right)\cdot 103^{4} + \left(59 a + 7\right)\cdot 103^{5} + \left(32 a + 30\right)\cdot 103^{6} + \left(73 a + 35\right)\cdot 103^{7} + \left(18 a + 38\right)\cdot 103^{8} + \left(58 a + 59\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 19 + 67\cdot 103 + 9\cdot 103^{2} + 94\cdot 103^{3} + 102\cdot 103^{4} + 26\cdot 103^{5} + 25\cdot 103^{6} + 36\cdot 103^{7} + 2\cdot 103^{8} + 70\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 64 a + 40 + \left(7 a + 57\right)\cdot 103 + \left(80 a + 32\right)\cdot 103^{2} + \left(54 a + 42\right)\cdot 103^{3} + \left(35 a + 45\right)\cdot 103^{4} + \left(86 a + 64\right)\cdot 103^{5} + \left(86 a + 54\right)\cdot 103^{6} + \left(20 a + 86\right)\cdot 103^{7} + \left(55 a + 67\right)\cdot 103^{8} + \left(56 a + 39\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 28 a + 93 + \left(80 a + 21\right)\cdot 103 + \left(47 a + 40\right)\cdot 103^{2} + \left(44 a + 53\right)\cdot 103^{3} + \left(18 a + 87\right)\cdot 103^{4} + \left(8 a + 38\right)\cdot 103^{5} + \left(18 a + 94\right)\cdot 103^{6} + \left(84 a + 79\right)\cdot 103^{7} + \left(87 a + 69\right)\cdot 103^{8} + \left(99 a + 77\right)\cdot 103^{9} +O(103^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(1,5)(4,8)$ | $-6$ |
| $9$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $2$ |
| $12$ | $2$ | $(2,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,5)(2,3)(4,8)$ | $0$ |
| $16$ | $3$ | $(2,6,7)$ | $0$ |
| $64$ | $3$ | $(2,6,7)(4,5,8)$ | $0$ |
| $12$ | $4$ | $(1,4,5,8)$ | $0$ |
| $36$ | $4$ | $(1,4,5,8)(2,3,6,7)$ | $-2$ |
| $36$ | $4$ | $(1,5)(2,3,6,7)(4,8)$ | $0$ |
| $72$ | $4$ | $(1,6,5,2)(3,4,7,8)$ | $0$ |
| $72$ | $4$ | $(1,4,5,8)(2,3)$ | $2$ |
| $144$ | $4$ | $(1,2,4,3)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,5)(2,7,6)(4,8)$ | $0$ |
| $96$ | $6$ | $(2,3)(4,8,5)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,4,6,5,7,8)$ | $0$ |
| $144$ | $8$ | $(1,3,4,6,5,7,8,2)$ | $0$ |
| $96$ | $12$ | $(1,4,5,8)(2,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.