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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 30 + \left(56 a + 21\right)\cdot 67 + \left(64 a + 36\right)\cdot 67^{2} + \left(37 a + 45\right)\cdot 67^{3} + \left(7 a + 6\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 24\cdot 67 + 44\cdot 67^{2} + 36\cdot 67^{3} + 30\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 60 a + 54 + \left(59 a + 40\right)\cdot 67 + \left(29 a + 19\right)\cdot 67^{2} + \left(23 a + 22\right)\cdot 67^{3} + \left(54 a + 52\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 26 + \left(7 a + 19\right)\cdot 67 + \left(37 a + 12\right)\cdot 67^{2} + \left(43 a + 19\right)\cdot 67^{3} + \left(12 a + 45\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 + 12\cdot 67 + 50\cdot 67^{2} + 11\cdot 67^{3} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 20 + \left(10 a + 15\right)\cdot 67 + \left(2 a + 38\right)\cdot 67^{2} + \left(29 a + 65\right)\cdot 67^{3} + \left(59 a + 65\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$16$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $15$ |
$2$ |
$(1,2)$ |
$0$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-2$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
