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 value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
