Basic invariants
Dimension: | $8$ |
Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Conductor: | \(2167393450849\)\(\medspace = 11^{4} \cdot 23^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.33860761.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T177 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $C_3^3:S_4$ |
Projective stem field: | Galois closure of 9.1.33860761.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 2x^{8} + 2x^{7} - 2x^{5} + 2x^{4} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 223 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 223 }$: \( x^{3} + 6x + 220 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 197\cdot 223 + 26\cdot 223^{2} + 153\cdot 223^{3} + 88\cdot 223^{4} + 9\cdot 223^{5} + 56\cdot 223^{6} + 187\cdot 223^{7} + 69\cdot 223^{8} + 168\cdot 223^{9} +O(223^{10})\)
$r_{ 2 }$ |
$=$ |
\( 43 + 192\cdot 223 + 139\cdot 223^{2} + 54\cdot 223^{3} + 53\cdot 223^{4} + 87\cdot 223^{5} + 103\cdot 223^{6} + 212\cdot 223^{7} + 170\cdot 223^{8} + 198\cdot 223^{9} +O(223^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 70 + 50\cdot 223 + 41\cdot 223^{2} + 141\cdot 223^{3} + 116\cdot 223^{4} + 48\cdot 223^{5} + 145\cdot 223^{6} + 84\cdot 223^{7} + 96\cdot 223^{8} + 177\cdot 223^{9} +O(223^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 27 a^{2} + 100 a + 14 + \left(152 a^{2} + 10 a + 131\right)\cdot 223 + \left(79 a^{2} + 143 a + 116\right)\cdot 223^{2} + \left(31 a^{2} + 78 a + 122\right)\cdot 223^{3} + \left(112 a^{2} + 3 a + 153\right)\cdot 223^{4} + \left(a^{2} + 183 a + 34\right)\cdot 223^{5} + \left(78 a^{2} + 55 a + 97\right)\cdot 223^{6} + \left(50 a^{2} + 36 a + 206\right)\cdot 223^{7} + \left(18 a^{2} + 143 a + 81\right)\cdot 223^{8} + \left(123 a^{2} + 147 a + 160\right)\cdot 223^{9} +O(223^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 39 a^{2} + 51 a + 62 + \left(75 a^{2} + 214 a + 46\right)\cdot 223 + \left(12 a^{2} + 109 a + 70\right)\cdot 223^{2} + \left(156 a^{2} + 115 a + 175\right)\cdot 223^{3} + \left(145 a^{2} + 130 a + 64\right)\cdot 223^{4} + \left(147 a^{2} + 58 a + 173\right)\cdot 223^{5} + \left(207 a^{2} + 209 a + 169\right)\cdot 223^{6} + \left(57 a^{2} + 203 a + 13\right)\cdot 223^{7} + \left(166 a^{2} + 197 a + 5\right)\cdot 223^{8} + \left(27 a^{2} + 37 a + 2\right)\cdot 223^{9} +O(223^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 41 a^{2} + 89 a + 146 + \left(204 a^{2} + 5 a + 32\right)\cdot 223 + \left(100 a^{2} + 101 a + 16\right)\cdot 223^{2} + \left(198 a^{2} + 189 a + 160\right)\cdot 223^{3} + \left(171 a^{2} + 56 a + 78\right)\cdot 223^{4} + \left(170 a^{2} + 184 a + 11\right)\cdot 223^{5} + \left(10 a^{2} + 164 a + 82\right)\cdot 223^{6} + \left(3 a^{2} + 132 a + 217\right)\cdot 223^{7} + \left(165 a^{2} + 149 a + 166\right)\cdot 223^{8} + \left(36 a^{2} + 124 a + 148\right)\cdot 223^{9} +O(223^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 157 a^{2} + 72 a + 88 + \left(218 a^{2} + 221 a + 174\right)\cdot 223 + \left(130 a^{2} + 192 a + 98\right)\cdot 223^{2} + \left(35 a^{2} + 28 a + 139\right)\cdot 223^{3} + \left(188 a^{2} + 89 a + 11\right)\cdot 223^{4} + \left(73 a^{2} + 204 a + 101\right)\cdot 223^{5} + \left(160 a^{2} + 180 a + 203\right)\cdot 223^{6} + \left(114 a^{2} + 205 a + 17\right)\cdot 223^{7} + \left(38 a^{2} + 104 a + 163\right)\cdot 223^{8} + \left(72 a^{2} + 37 a + 179\right)\cdot 223^{9} +O(223^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 201 a^{2} + 145 a + 117 + \left(182 a^{2} + 193 a + 170\right)\cdot 223 + \left(66 a^{2} + 187 a + 102\right)\cdot 223^{2} + \left(23 a^{2} + 135 a + 128\right)\cdot 223^{3} + \left(50 a^{2} + 91 a + 37\right)\cdot 223^{4} + \left(a^{2} + 92 a + 2\right)\cdot 223^{5} + \left(222 a^{2} + 175 a + 35\right)\cdot 223^{6} + \left(22 a^{2} + 63 a + 74\right)\cdot 223^{7} + \left(145 a^{2} + 220 a + 87\right)\cdot 223^{8} + \left(125 a^{2} + 174 a + 58\right)\cdot 223^{9} +O(223^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 204 a^{2} + 212 a + 129 + \left(58 a^{2} + 23 a + 120\right)\cdot 223 + \left(55 a^{2} + 157 a + 56\right)\cdot 223^{2} + \left(a^{2} + 120 a + 40\right)\cdot 223^{3} + \left(a^{2} + 74 a + 64\right)\cdot 223^{4} + \left(51 a^{2} + 169 a + 201\right)\cdot 223^{5} + \left(213 a^{2} + 105 a + 222\right)\cdot 223^{6} + \left(196 a^{2} + 26 a + 100\right)\cdot 223^{7} + \left(135 a^{2} + 76 a + 50\right)\cdot 223^{8} + \left(60 a^{2} + 146 a + 21\right)\cdot 223^{9} +O(223^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
$1$ | $1$ | $()$ | $8$ |
$27$ | $2$ | $(4,5)(6,8)$ | $0$ |
$54$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $0$ |
$6$ | $3$ | $(6,9,8)$ | $-4$ |
$8$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $-1$ |
$12$ | $3$ | $(4,7,5)(6,9,8)$ | $2$ |
$72$ | $3$ | $(1,4,6)(2,5,8)(3,7,9)$ | $-1$ |
$54$ | $4$ | $(4,6,5,8)(7,9)$ | $0$ |
$54$ | $6$ | $(1,2)(4,5)(6,8,9)$ | $0$ |
$108$ | $6$ | $(1,2)(4,6,7,9,5,8)$ | $0$ |
$72$ | $9$ | $(1,4,6,2,5,8,3,7,9)$ | $-1$ |
$72$ | $9$ | $(1,4,6,3,7,9,2,5,8)$ | $2$ |
$54$ | $12$ | $(1,5,2,4)(3,7)(6,9,8)$ | $0$ |
$54$ | $12$ | $(1,5,2,4)(3,7)(6,8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.