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:
| $r_{ 1 }$ |
$=$ |
$ 120 a + 144 + \left(128 a + 120\right)\cdot 197 + \left(69 a + 168\right)\cdot 197^{2} + \left(190 a + 107\right)\cdot 197^{3} + \left(32 a + 172\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 142 + 13\cdot 197 + 150\cdot 197^{2} + 8\cdot 197^{3} + 167\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 106 + 75\cdot 197 + 62\cdot 197^{2} + 194\cdot 197^{3} + 194\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 52 + \left(196 a + 180\right)\cdot 197 + \left(143 a + 42\right)\cdot 197^{2} + \left(91 a + 177\right)\cdot 197^{3} + \left(45 a + 83\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 169 a + 192 + 147\cdot 197 + \left(53 a + 172\right)\cdot 197^{2} + \left(105 a + 97\right)\cdot 197^{3} + \left(151 a + 22\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 77 a + 153 + \left(68 a + 52\right)\cdot 197 + \left(127 a + 191\right)\cdot 197^{2} + \left(6 a + 4\right)\cdot 197^{3} + \left(164 a + 147\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $15$ |
$2$ |
$(1,2)$ |
$1$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
