Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 37 a + 50 + \left(47 a + 15\right)\cdot 67 + \left(4 a + 33\right)\cdot 67^{2} + \left(14 a + 62\right)\cdot 67^{3} + \left(13 a + 9\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 + 32\cdot 67 + 51\cdot 67^{2} + 25\cdot 67^{3} + 27\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 a + 57 + \left(2 a + 7\right)\cdot 67 + \left(23 a + 41\right)\cdot 67^{2} + \left(45 a + 41\right)\cdot 67^{3} + \left(12 a + 4\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 59 + 5\cdot 67 + 7\cdot 67^{2} + 25\cdot 67^{3} + 33\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a + 52 + \left(64 a + 36\right)\cdot 67 + \left(43 a + 63\right)\cdot 67^{2} + \left(21 a + 65\right)\cdot 67^{3} + \left(54 a + 9\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a + 64 + \left(19 a + 34\right)\cdot 67 + \left(62 a + 4\right)\cdot 67^{2} + \left(52 a + 47\right)\cdot 67^{3} + \left(53 a + 48\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,6)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(3,5)$ |
$0$ |
| $9$ |
$2$ |
$(3,5)(4,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,4,6)$ |
$-2$ |
| $4$ |
$3$ |
$(1,4,6)(2,3,5)$ |
$1$ |
| $18$ |
$4$ |
$(1,2)(3,6,5,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,4,5,6,2)$ |
$1$ |
| $12$ |
$6$ |
$(1,4,6)(3,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
