Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 197 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 197 }$: $ x^{2} + 192 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 a + 18 + \left(13 a + 185\right)\cdot 197 + \left(157 a + 84\right)\cdot 197^{2} + \left(72 a + 146\right)\cdot 197^{3} + \left(125 a + 111\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 193 a + 17 + \left(152 a + 10\right)\cdot 197 + \left(114 a + 144\right)\cdot 197^{2} + \left(68 a + 131\right)\cdot 197^{3} + \left(182 a + 44\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 153 + 2\cdot 197 + 71\cdot 197^{2} + 5\cdot 197^{3} + 137\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 76 + 180\cdot 197 + 132\cdot 197^{2} + 177\cdot 197^{3} + 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 174 a + 133 + \left(183 a + 30\right)\cdot 197 + \left(39 a + 69\right)\cdot 197^{2} + \left(124 a + 156\right)\cdot 197^{3} + \left(71 a + 74\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 + 191\cdot 197 + 114\cdot 197^{2} + 7\cdot 197^{3} + 121\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 4 a + 194 + \left(44 a + 187\right)\cdot 197 + \left(82 a + 170\right)\cdot 197^{2} + \left(128 a + 162\right)\cdot 197^{3} + \left(14 a + 99\right)\cdot 197^{4} +O\left(197^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,2)$ |
$-4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
