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:
| $r_{ 1 }$ |
$=$ |
$ 58 a + 3 + \left(38 a + 63\right)\cdot 67 + \left(5 a + 26\right)\cdot 67^{2} + \left(47 a + 2\right)\cdot 67^{3} + \left(42 a + 54\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 34 + \left(28 a + 26\right)\cdot 67 + \left(61 a + 10\right)\cdot 67^{2} + \left(19 a + 51\right)\cdot 67^{3} + \left(24 a + 43\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 + 33\cdot 67 + 21\cdot 67^{2} + 54\cdot 67^{3} + 2\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 + 31\cdot 67 + 9\cdot 67^{2} + 27\cdot 67^{3} + 41\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 a + 41 + \left(45 a + 27\right)\cdot 67 + \left(54 a + 13\right)\cdot 67^{2} + \left(34 a + 24\right)\cdot 67^{3} + \left(53 a + 40\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 8 a + 9 + \left(21 a + 18\right)\cdot 67 + \left(12 a + 52\right)\cdot 67^{2} + \left(32 a + 41\right)\cdot 67^{3} + \left(13 a + 18\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$9$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
