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:
| $r_{ 1 }$ |
$=$ |
$ 4 a + 10 + \left(38 a + 30\right)\cdot 67 + \left(38 a + 23\right)\cdot 67^{2} + \left(9 a + 51\right)\cdot 67^{3} + \left(19 a + 23\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 a + 60 + \left(2 a + 65\right)\cdot 67 + \left(17 a + 22\right)\cdot 67^{2} + \left(53 a + 27\right)\cdot 67^{3} + \left(55 a + 58\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 66 + \left(15 a + 25\right)\cdot 67 + \left(61 a + 49\right)\cdot 67^{2} + \left(52 a + 49\right)\cdot 67^{3} + \left(31 a + 63\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 63 a + 26 + \left(28 a + 44\right)\cdot 67 + \left(28 a + 5\right)\cdot 67^{2} + \left(57 a + 51\right)\cdot 67^{3} + \left(47 a + 23\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a + 55 + \left(64 a + 27\right)\cdot 67 + \left(49 a + 21\right)\cdot 67^{2} + \left(13 a + 22\right)\cdot 67^{3} + \left(11 a + 27\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 a + 51 + \left(51 a + 6\right)\cdot 67 + \left(5 a + 11\right)\cdot 67^{2} + \left(14 a + 66\right)\cdot 67^{3} + \left(35 a + 3\right)\cdot 67^{4} +O\left(67^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$10$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $15$ |
$2$ |
$(1,2)$ |
$2$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
