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:
| $r_{ 1 }$ |
$=$ |
$ 18 + 3\cdot 67 + 26\cdot 67^{2} + 39\cdot 67^{3} + 47\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 44\cdot 67 + 8\cdot 67^{2} + 28\cdot 67^{3} + 20\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 62 + \left(55 a + 41\right)\cdot 67 + \left(48 a + 66\right)\cdot 67^{2} + \left(13 a + 23\right)\cdot 67^{3} + \left(42 a + 36\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 a + 3 + \left(11 a + 59\right)\cdot 67 + \left(18 a + 5\right)\cdot 67^{2} + \left(53 a + 30\right)\cdot 67^{3} + \left(24 a + 57\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 53\cdot 67 + 33\cdot 67^{2} + 27\cdot 67^{3} + 10\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 66\cdot 67 + 59\cdot 67^{2} + 51\cdot 67^{3} + 28\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-1$ |
| $15$ |
$2$ |
$(1,2)$ |
$3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
