Basic invariants
Dimension: | $2$ |
Group: | $S_3 \times C_5$ |
Conductor: | \(9801\)\(\medspace = 3^{4} \cdot 11^{2} \) |
Artin stem field: | Galois closure of 15.5.120373254183671877620331.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $S_3 \times C_5$ |
Parity: | odd |
Determinant: | 1.11.10t1.a.a |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.891.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{15} + 6 x^{13} - 19 x^{12} - 45 x^{11} + 87 x^{10} - 123 x^{9} + 117 x^{8} - 228 x^{7} - 555 x^{6} + 2619 x^{5} + 2145 x^{4} - 4231 x^{3} - 3645 x^{2} + 1209 x + 2069 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 223 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 223 }$: \( x^{5} + x + 220 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 30 a^{4} + 39 a^{3} + 41 a^{2} + 130 a + 127 + \left(119 a^{4} + 203 a^{3} + 59 a^{2} + 105 a + 58\right)\cdot 223 + \left(73 a^{4} + 218 a^{3} + 202 a^{2} + 50 a + 203\right)\cdot 223^{2} + \left(71 a^{4} + 50 a^{3} + 115 a^{2} + 63 a + 151\right)\cdot 223^{3} + \left(200 a^{4} + 164 a^{3} + 71 a^{2} + 194 a + 172\right)\cdot 223^{4} + \left(58 a^{4} + 129 a^{3} + 175 a^{2} + 44 a + 50\right)\cdot 223^{5} +O(223^{6})\)
$r_{ 2 }$ |
$=$ |
\( 44 a^{4} + 33 a^{3} + 87 a^{2} + 189 a + 73 + \left(152 a^{4} + 169 a^{3} + 198 a^{2} + 184 a + 199\right)\cdot 223 + \left(109 a^{4} + 14 a^{3} + 41 a^{2} + 32 a + 151\right)\cdot 223^{2} + \left(131 a^{4} + 98 a^{3} + 103 a^{2} + 200 a + 143\right)\cdot 223^{3} + \left(153 a^{4} + 155 a^{3} + 175 a^{2} + 138 a + 77\right)\cdot 223^{4} + \left(131 a^{4} + 183 a^{3} + 124 a^{2} + 110 a + 215\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 44 a^{4} + 120 a^{3} + 151 a^{2} + 149 a + 162 + \left(115 a^{4} + 127 a^{3} + 179 a^{2} + 24 a + 6\right)\cdot 223 + \left(73 a^{4} + 33 a^{3} + 9 a^{2} + 99 a + 207\right)\cdot 223^{2} + \left(191 a^{4} + 61 a^{3} + 150 a^{2} + 194 a + 108\right)\cdot 223^{3} + \left(151 a^{4} + 55 a^{3} + 17 a^{2} + 55 a + 20\right)\cdot 223^{4} + \left(211 a^{4} + 96 a^{3} + 81 a^{2} + 49 a + 11\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 46 a^{4} + 47 a^{3} + 27 a^{2} + 193 a + 119 + \left(99 a^{4} + 54 a^{3} + 95 a^{2} + 28 a + 38\right)\cdot 223 + \left(118 a^{4} + 133 a^{3} + 83 a^{2} + 213 a + 109\right)\cdot 223^{2} + \left(120 a^{4} + 85 a^{3} + 63 a^{2} + 116 a + 141\right)\cdot 223^{3} + \left(106 a^{4} + 195 a^{3} + 112 a^{2} + 4 a + 162\right)\cdot 223^{4} + \left(190 a^{4} + 128 a^{3} + 159 a^{2} + 185 a + 38\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 69 a^{4} + 23 a^{3} + 137 a^{2} + 74 a + 182 + \left(190 a^{4} + 27 a^{3} + 69 a^{2} + 175 a + 66\right)\cdot 223 + \left(159 a^{4} + 65 a^{3} + 152 a^{2} + 82 a + 142\right)\cdot 223^{2} + \left(101 a^{4} + 75 a^{3} + 201 a^{2} + 167 a + 215\right)\cdot 223^{3} + \left(197 a^{4} + 70 a^{3} + 13 a^{2} + 110 a + 56\right)\cdot 223^{4} + \left(192 a^{4} + 153 a^{3} + 222 a^{2} + 123 a + 85\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 72 a^{4} + 153 a^{3} + 42 a^{2} + 101 a + 116 + \left(107 a^{4} + 3 a^{3} + 208 a^{2} + 62 a + 4\right)\cdot 223 + \left(40 a^{4} + 143 a^{3} + 176 a^{2} + 126 a + 43\right)\cdot 223^{2} + \left(114 a^{4} + 84 a^{3} + 193 a^{2} + 85 a + 186\right)\cdot 223^{3} + \left(93 a^{4} + 46 a^{3} + 107 a^{2} + 157 a + 131\right)\cdot 223^{4} + \left(132 a^{4} + 126 a^{3} + 128 a^{2} + 220 a + 109\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 7 }$ |
$=$ |
\( 107 a^{4} + 37 a^{3} + 94 a^{2} + 156 a + 34 + \left(186 a^{4} + 50 a^{3} + 39 a^{2} + 198 a + 19\right)\cdot 223 + \left(72 a^{4} + 65 a^{3} + 4 a^{2} + 63 a + 28\right)\cdot 223^{2} + \left(200 a^{4} + 40 a^{3} + 149 a^{2} + 160 a + 116\right)\cdot 223^{3} + \left(198 a^{4} + 21 a^{3} + 162 a^{2} + 149 a + 13\right)\cdot 223^{4} + \left(181 a^{4} + 136 a^{3} + 192 a^{2} + 114 a + 121\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 8 }$ |
$=$ |
\( 107 a^{4} + 216 a^{3} + 136 a^{2} + 65 a + 168 + \left(32 a^{4} + 69 a^{3} + 192 a^{2} + 127 a + 192\right)\cdot 223 + \left(178 a^{4} + 41 a^{3} + 88 a^{2} + 46 a + 72\right)\cdot 223^{2} + \left(200 a^{4} + 107 a^{3} + 199 a^{2} + 182 a + 65\right)\cdot 223^{3} + \left(24 a^{4} + 62 a^{3} + 193 a^{2} + 80 a + 19\right)\cdot 223^{4} + \left(132 a^{4} + 127 a^{3} + 34 a^{2} + 212 a + 171\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 9 }$ |
$=$ |
\( 147 a^{4} + 137 a^{3} + 155 a^{2} + 123 a + 200 + \left(4 a^{4} + 188 a^{3} + 68 a^{2} + 88 a + 125\right)\cdot 223 + \left(31 a^{4} + 93 a^{3} + 160 a^{2} + 182 a + 133\right)\cdot 223^{2} + \left(31 a^{4} + 86 a^{3} + 43 a^{2} + 42 a + 152\right)\cdot 223^{3} + \left(139 a^{4} + 86 a^{3} + 39 a^{2} + 24 a + 110\right)\cdot 223^{4} + \left(196 a^{4} + 187 a^{3} + 111 a^{2} + 216 a + 133\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 10 }$ |
$=$ |
\( 159 a^{4} + 11 a^{3} + 50 a^{2} + 61 a + 141 + \left(112 a^{4} + 189 a^{3} + 191 a^{2} + 77 a + 142\right)\cdot 223 + \left(23 a^{4} + 32 a^{3} + 160 a^{2} + 189 a + 118\right)\cdot 223^{2} + \left(190 a^{4} + 155 a^{3} + 141 a^{2} + 10 a + 68\right)\cdot 223^{3} + \left(183 a^{4} + 56 a^{3} + 112 a^{2} + 17 a + 70\right)\cdot 223^{4} + \left(198 a^{4} + 164 a^{3} + 174 a^{2} + 37 a + 207\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 11 }$ |
$=$ |
\( 168 a^{4} + 150 a^{3} + 43 a^{2} + 193 a + 83 + \left(55 a^{4} + 201 a^{3} + 83 a^{2} + 153 a + 211\right)\cdot 223 + \left(128 a^{4} + 213 a^{3} + 213 a^{2} + 9 a + 32\right)\cdot 223^{2} + \left(153 a^{4} + 113 a^{3} + 109 a^{2} + 200 a + 72\right)\cdot 223^{3} + \left(92 a^{4} + 47 a^{3} + 141 a^{2} + 113 a + 207\right)\cdot 223^{4} + \left(160 a^{4} + 37 a^{3} + 160 a^{2} + 15 a + 59\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 12 }$ |
$=$ |
\( 180 a^{4} + 219 a^{3} + 37 a^{2} + 97 a + 137 + \left(77 a^{4} + 186 a^{3} + 62 a^{2} + 18 a + 110\right)\cdot 223 + \left(21 a^{4} + 148 a^{3} + 196 a^{2} + 210 a + 31\right)\cdot 223^{2} + \left(55 a^{4} + 183 a^{3} + 104 a^{2} + 29 a + 89\right)\cdot 223^{3} + \left(14 a^{4} + 103 a^{3} + 139 a^{2} + 125 a + 133\right)\cdot 223^{4} + \left(115 a^{4} + 154 a^{3} + 13 a^{2} + 196 a + 67\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 13 }$ |
$=$ |
\( 199 a^{4} + 193 a^{3} + 47 a^{2} + 198 a + 173 + \left(129 a^{4} + 55 a^{3} + 140 a^{2} + 83 a + 111\right)\cdot 223 + \left(150 a^{4} + 107 a^{3} + 48 a^{2} + 172 a + 86\right)\cdot 223^{2} + \left(102 a^{4} + 121 a^{3} + 83 a^{2} + 207 a + 132\right)\cdot 223^{3} + \left(35 a^{4} + 73 a^{3} + 86 a^{2} + 78 a + 174\right)\cdot 223^{4} + \left(186 a^{4} + 216 a^{3} + 127 a^{2} + 59 a + 18\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 14 }$ |
$=$ |
\( 203 a^{4} + 133 a^{3} + 25 a^{2} + 99 a + 111 + \left(200 a^{4} + 39 a^{3} + 126 a^{2} + 114 a + 104\right)\cdot 223 + \left(221 a^{4} + 82 a^{3} + 164 a^{2} + 174 a + 152\right)\cdot 223^{2} + \left(151 a^{4} + 40 a^{3} + 212 a^{2} + 43 a + 204\right)\cdot 223^{3} + \left(35 a^{4} + 94 a^{3} + 118 a^{2} + 88 a + 27\right)\cdot 223^{4} + \left(48 a^{4} + 133 a^{3} + 14 a^{2} + 114 a + 193\right)\cdot 223^{5} +O(223^{6})\)
| $r_{ 15 }$ |
$=$ |
\( 209 a^{4} + 50 a^{3} + 43 a^{2} + 179 a + 181 + \left(199 a^{4} + 217 a^{3} + 70 a^{2} + 116 a + 167\right)\cdot 223 + \left(157 a^{4} + 166 a^{3} + 80 a^{2} + 130 a + 47\right)\cdot 223^{2} + \left(190 a^{4} + 33 a^{3} + 134 a^{2} + 78 a + 158\right)\cdot 223^{3} + \left(155 a^{4} + 105 a^{3} + 67 a^{2} + 221 a + 181\right)\cdot 223^{4} + \left(92 a^{4} + 32 a^{3} + 63 a^{2} + 83 a + 77\right)\cdot 223^{5} +O(223^{6})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 15 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 15 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,9)(2,6)(8,10)(11,15)(13,14)$ | $0$ |
$2$ | $3$ | $(1,9,4)(2,7,6)(3,13,14)(5,15,11)(8,12,10)$ | $-1$ |
$1$ | $5$ | $(1,10,15,13,6)(2,9,8,11,14)(3,7,4,12,5)$ | $2 \zeta_{5}^{3}$ |
$1$ | $5$ | $(1,15,6,10,13)(2,8,14,9,11)(3,4,5,7,12)$ | $2 \zeta_{5}$ |
$1$ | $5$ | $(1,13,10,6,15)(2,11,9,14,8)(3,12,7,5,4)$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ |
$1$ | $5$ | $(1,6,13,15,10)(2,14,11,8,9)(3,5,12,4,7)$ | $2 \zeta_{5}^{2}$ |
$3$ | $10$ | $(1,14,10,2,15,9,13,8,6,11)(3,12,7,5,4)$ | $0$ |
$3$ | $10$ | $(1,2,13,11,10,9,6,14,15,8)(3,5,12,4,7)$ | $0$ |
$3$ | $10$ | $(1,8,15,14,6,9,10,11,13,2)(3,7,4,12,5)$ | $0$ |
$3$ | $10$ | $(1,11,6,8,13,9,15,2,10,14)(3,4,5,7,12)$ | $0$ |
$2$ | $15$ | $(1,3,8,6,5,9,13,12,2,15,4,14,10,7,11)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$2$ | $15$ | $(1,8,5,13,2,4,10,11,3,6,9,12,15,14,7)$ | $-\zeta_{5}^{3}$ |
$2$ | $15$ | $(1,5,2,10,3,9,15,7,8,13,4,11,6,12,14)$ | $-\zeta_{5}$ |
$2$ | $15$ | $(1,2,3,15,8,4,6,14,5,10,9,7,13,11,12)$ | $-\zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.