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:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 26 + \left(50 a + 166\right)\cdot 199 + \left(88 a + 61\right)\cdot 199^{2} + \left(103 a + 86\right)\cdot 199^{3} + \left(144 a + 67\right)\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 184 a + 116 + \left(148 a + 53\right)\cdot 199 + \left(110 a + 143\right)\cdot 199^{2} + \left(95 a + 21\right)\cdot 199^{3} + \left(54 a + 35\right)\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 183 + 98\cdot 199 + 171\cdot 199^{2} + 109\cdot 199^{3} + 13\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 138\cdot 199 + 3\cdot 199^{2} + 110\cdot 199^{3} + 102\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 189 a + 14 + \left(6 a + 24\right)\cdot 199 + \left(152 a + 162\right)\cdot 199^{2} + \left(151 a + 74\right)\cdot 199^{3} + \left(186 a + 35\right)\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 153 + \left(192 a + 75\right)\cdot 199 + \left(46 a + 72\right)\cdot 199^{2} + \left(47 a + 37\right)\cdot 199^{3} + \left(12 a + 9\right)\cdot 199^{4} +O\left(199^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 40\cdot 199 + 181\cdot 199^{2} + 156\cdot 199^{3} + 134\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 value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
