Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(241\!\cdots\!528\)\(\medspace = 2^{12} \cdot 23^{9} \cdot 41^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.77147686244.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.943.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.77147686244.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 2x^{6} - 8x^{5} + 22x^{4} + 25x^{3} + 36x^{2} + 23 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$:
\( x^{2} + 192x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 127 + 43\cdot 193 + 15\cdot 193^{2} + 46\cdot 193^{3} + 21\cdot 193^{4} + 131\cdot 193^{5} + 143\cdot 193^{6} + 137\cdot 193^{7} + 131\cdot 193^{8} + 157\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 104 a + 162 + \left(134 a + 57\right)\cdot 193 + \left(96 a + 162\right)\cdot 193^{2} + \left(73 a + 184\right)\cdot 193^{3} + \left(99 a + 40\right)\cdot 193^{4} + \left(174 a + 37\right)\cdot 193^{5} + \left(60 a + 88\right)\cdot 193^{6} + \left(40 a + 70\right)\cdot 193^{7} + \left(167 a + 12\right)\cdot 193^{8} + \left(111 a + 94\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 161 a + 82 + \left(75 a + 68\right)\cdot 193 + \left(175 a + 126\right)\cdot 193^{2} + \left(187 a + 169\right)\cdot 193^{3} + \left(7 a + 62\right)\cdot 193^{4} + \left(172 a + 38\right)\cdot 193^{5} + \left(10 a + 128\right)\cdot 193^{6} + \left(89 a + 50\right)\cdot 193^{7} + \left(114 a + 81\right)\cdot 193^{8} + \left(50 a + 109\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 32 a + 50 + \left(117 a + 176\right)\cdot 193 + \left(17 a + 32\right)\cdot 193^{2} + \left(5 a + 182\right)\cdot 193^{3} + \left(185 a + 75\right)\cdot 193^{4} + \left(20 a + 9\right)\cdot 193^{5} + \left(182 a + 160\right)\cdot 193^{6} + \left(103 a + 128\right)\cdot 193^{7} + \left(78 a + 106\right)\cdot 193^{8} + \left(142 a + 45\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a + 160 + \left(187 a + 84\right)\cdot 193 + \left(91 a + 7\right)\cdot 193^{2} + \left(180 a + 154\right)\cdot 193^{3} + \left(184 a + 159\right)\cdot 193^{4} + \left(107 a + 122\right)\cdot 193^{5} + \left(183 a + 141\right)\cdot 193^{6} + \left(157 a + 166\right)\cdot 193^{7} + \left(117 a + 174\right)\cdot 193^{8} + \left(18 a + 1\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 145 + 180\cdot 193 + 4\cdot 193^{2} + 17\cdot 193^{3} + 180\cdot 193^{4} + 81\cdot 193^{5} + 111\cdot 193^{6} + 26\cdot 193^{7} + 184\cdot 193^{8} + 35\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 89 a + 73 + \left(58 a + 88\right)\cdot 193 + \left(96 a + 124\right)\cdot 193^{2} + \left(119 a + 161\right)\cdot 193^{3} + \left(93 a + 66\right)\cdot 193^{4} + \left(18 a + 112\right)\cdot 193^{5} + \left(132 a + 167\right)\cdot 193^{6} + \left(152 a + 49\right)\cdot 193^{7} + \left(25 a + 139\right)\cdot 193^{8} + \left(81 a + 38\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 186 a + 167 + \left(5 a + 71\right)\cdot 193 + \left(101 a + 105\right)\cdot 193^{2} + \left(12 a + 49\right)\cdot 193^{3} + \left(8 a + 164\right)\cdot 193^{4} + \left(85 a + 45\right)\cdot 193^{5} + \left(9 a + 24\right)\cdot 193^{6} + \left(35 a + 141\right)\cdot 193^{7} + \left(75 a + 134\right)\cdot 193^{8} + \left(174 a + 95\right)\cdot 193^{9} +O(193^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,4)(3,7)$ | $-6$ |
| $9$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $2$ |
| $12$ | $2$ | $(1,5)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,5)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,5)(2,4)(3,7)$ | $0$ |
| $16$ | $3$ | $(1,6,8)$ | $0$ |
| $64$ | $3$ | $(1,6,8)(3,4,7)$ | $0$ |
| $12$ | $4$ | $(2,3,4,7)$ | $0$ |
| $36$ | $4$ | $(1,5,6,8)(2,3,4,7)$ | $-2$ |
| $36$ | $4$ | $(1,5,6,8)(2,4)(3,7)$ | $0$ |
| $72$ | $4$ | $(1,2,6,4)(3,8,7,5)$ | $0$ |
| $72$ | $4$ | $(1,5)(2,3,4,7)$ | $2$ |
| $144$ | $4$ | $(1,3,5,2)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,8,6)(2,4)(3,7)$ | $0$ |
| $96$ | $6$ | $(1,5)(3,7,4)$ | $0$ |
| $192$ | $6$ | $(1,3,6,4,8,7)(2,5)$ | $0$ |
| $144$ | $8$ | $(1,2,5,3,6,4,8,7)$ | $0$ |
| $96$ | $12$ | $(1,6,8)(2,3,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.