Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$: $ x^{2} + 193 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 a + 134 + \left(52 a + 128\right)\cdot 199 + \left(5 a + 149\right)\cdot 199^{2} + \left(193 a + 4\right)\cdot 199^{3} + \left(155 a + 97\right)\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 56 + \left(120 a + 85\right)\cdot 199 + \left(42 a + 175\right)\cdot 199^{2} + \left(186 a + 151\right)\cdot 199^{3} + \left(145 a + 94\right)\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 158 + 96\cdot 199 + 71\cdot 199^{2} + 132\cdot 199^{3} + 32\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 156 a + 193 + 146 a\cdot 199 + \left(193 a + 129\right)\cdot 199^{2} + \left(5 a + 162\right)\cdot 199^{3} + \left(43 a + 43\right)\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 172 a + 19 + \left(78 a + 182\right)\cdot 199 + \left(156 a + 111\right)\cdot 199^{2} + \left(12 a + 32\right)\cdot 199^{3} + \left(53 a + 187\right)\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 114 + 196\cdot 199 + 159\cdot 199^{2} + 66\cdot 199^{3} + 105\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 123 + 105\cdot 199 + 197\cdot 199^{2} + 45\cdot 199^{3} + 36\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$21$ |
| $21$ |
$2$ |
$(1,2)$ |
$1$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
