Basic invariants
Dimension: | $18$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(556\!\cdots\!424\)\(\medspace = 2^{53} \cdot 3^{31} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.1934917632.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 36T1758 |
Parity: | odd |
Determinant: | 1.24.2t1.b.a |
Projective image: | $S_4\wr C_2$ |
Projective stem field: | Galois closure of 8.2.1934917632.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 4x^{6} - 8x^{5} + 7x^{4} - 14x^{3} + 4x^{2} - 8x - 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$: \( x^{2} + 177x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 128 + 172\cdot 181 + 154\cdot 181^{2} + 148\cdot 181^{3} + 99\cdot 181^{4} + 72\cdot 181^{5} + 136\cdot 181^{6} + 114\cdot 181^{7} + 95\cdot 181^{8} + 13\cdot 181^{9} +O(181^{10})\)
$r_{ 2 }$ |
$=$ |
\( 150 + 37\cdot 181 + 120\cdot 181^{2} + 3\cdot 181^{3} + 110\cdot 181^{4} + 14\cdot 181^{5} + 149\cdot 181^{6} + 47\cdot 181^{7} + 103\cdot 181^{8} + 78\cdot 181^{9} +O(181^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 89 a + 53 + \left(16 a + 151\right)\cdot 181 + \left(115 a + 53\right)\cdot 181^{2} + \left(137 a + 179\right)\cdot 181^{3} + \left(151 a + 50\right)\cdot 181^{4} + \left(166 a + 108\right)\cdot 181^{5} + \left(178 a + 105\right)\cdot 181^{6} + \left(146 a + 134\right)\cdot 181^{7} + \left(87 a + 83\right)\cdot 181^{8} + \left(77 a + 16\right)\cdot 181^{9} +O(181^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 47 a + 21 + \left(27 a + 117\right)\cdot 181 + \left(4 a + 21\right)\cdot 181^{2} + \left(159 a + 52\right)\cdot 181^{3} + \left(157 a + 28\right)\cdot 181^{4} + \left(80 a + 133\right)\cdot 181^{5} + \left(17 a + 83\right)\cdot 181^{6} + \left(130 a + 65\right)\cdot 181^{7} + \left(84 a + 109\right)\cdot 181^{8} + \left(102 a + 47\right)\cdot 181^{9} +O(181^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 132 a + 157 + \left(159 a + 167\right)\cdot 181 + \left(56 a + 169\right)\cdot 181^{2} + \left(156 a + 50\right)\cdot 181^{3} + \left(112 a + 102\right)\cdot 181^{4} + \left(40 a + 73\right)\cdot 181^{5} + \left(135 a + 144\right)\cdot 181^{6} + \left(91 a + 51\right)\cdot 181^{7} + \left(46 a + 87\right)\cdot 181^{8} + \left(66 a + 140\right)\cdot 181^{9} +O(181^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 49 a + 142 + \left(21 a + 131\right)\cdot 181 + \left(124 a + 56\right)\cdot 181^{2} + \left(24 a + 76\right)\cdot 181^{3} + \left(68 a + 35\right)\cdot 181^{4} + \left(140 a + 123\right)\cdot 181^{5} + \left(45 a + 101\right)\cdot 181^{6} + \left(89 a + 102\right)\cdot 181^{7} + 134 a\cdot 181^{8} + \left(114 a + 178\right)\cdot 181^{9} +O(181^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 92 a + 47 + \left(164 a + 128\right)\cdot 181 + \left(65 a + 135\right)\cdot 181^{2} + \left(43 a + 71\right)\cdot 181^{3} + \left(29 a + 158\right)\cdot 181^{4} + \left(14 a + 80\right)\cdot 181^{5} + \left(2 a + 111\right)\cdot 181^{6} + 34 a\cdot 181^{7} + \left(93 a + 107\right)\cdot 181^{8} + \left(103 a + 57\right)\cdot 181^{9} +O(181^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 134 a + 28 + \left(153 a + 179\right)\cdot 181 + \left(176 a + 10\right)\cdot 181^{2} + \left(21 a + 141\right)\cdot 181^{3} + \left(23 a + 138\right)\cdot 181^{4} + \left(100 a + 117\right)\cdot 181^{5} + \left(163 a + 72\right)\cdot 181^{6} + \left(50 a + 25\right)\cdot 181^{7} + \left(96 a + 137\right)\cdot 181^{8} + \left(78 a + 10\right)\cdot 181^{9} +O(181^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $18$ |
$6$ | $2$ | $(3,7)(4,8)$ | $-6$ |
$9$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $2$ |
$12$ | $2$ | $(1,2)$ | $0$ |
$24$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
$36$ | $2$ | $(1,2)(3,8)$ | $-2$ |
$36$ | $2$ | $(1,2)(3,7)(4,8)$ | $0$ |
$16$ | $3$ | $(2,6,5)$ | $0$ |
$64$ | $3$ | $(2,6,5)(4,8,7)$ | $0$ |
$12$ | $4$ | $(3,4,7,8)$ | $0$ |
$36$ | $4$ | $(1,2,5,6)(3,4,7,8)$ | $-2$ |
$36$ | $4$ | $(1,2,5,6)(3,7)(4,8)$ | $0$ |
$72$ | $4$ | $(1,3,5,7)(2,4,6,8)$ | $0$ |
$72$ | $4$ | $(1,2)(3,4,7,8)$ | $2$ |
$144$ | $4$ | $(1,3,2,8)(4,5)(6,7)$ | $0$ |
$48$ | $6$ | $(2,5,6)(3,7)(4,8)$ | $0$ |
$96$ | $6$ | $(2,6,5)(3,4)$ | $0$ |
$192$ | $6$ | $(1,3)(2,8,6,7,5,4)$ | $0$ |
$144$ | $8$ | $(1,3,2,4,5,7,6,8)$ | $0$ |
$96$ | $12$ | $(2,6,5)(3,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.