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:
| $r_{ 1 }$ |
$=$ |
$ 146 + 125\cdot 197 + 105\cdot 197^{2} + 138\cdot 197^{3} + 42\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 157 a + 172 + \left(73 a + 8\right)\cdot 197 + \left(32 a + 55\right)\cdot 197^{2} + \left(76 a + 195\right)\cdot 197^{3} + \left(3 a + 44\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 160 + 167\cdot 197 + 22\cdot 197^{2} + 12\cdot 197^{3} + 73\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 98 + 105\cdot 197 + 73\cdot 197^{2} + 145\cdot 197^{3} + 126\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 165 + 85\cdot 197 + 22\cdot 197^{2} + 57\cdot 197^{3} + 50\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 a + 169 + \left(123 a + 23\right)\cdot 197 + \left(164 a + 143\right)\cdot 197^{2} + \left(120 a + 149\right)\cdot 197^{3} + \left(193 a + 182\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 + 73\cdot 197 + 168\cdot 197^{2} + 89\cdot 197^{3} + 70\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.
