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:
| $r_{ 1 }$ |
$=$ |
$ 9 a + 40 + \left(64 a + 53\right)\cdot 67 + \left(31 a + 22\right)\cdot 67^{2} + \left(48 a + 45\right)\cdot 67^{3} + \left(62 a + 66\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 47\cdot 67 + 24\cdot 67^{2} + 15\cdot 67^{3} + 66\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 a + 35 + \left(45 a + 64\right)\cdot 67 + \left(32 a + 29\right)\cdot 67^{2} + \left(32 a + 17\right)\cdot 67^{3} + \left(46 a + 18\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 a + 46 + \left(21 a + 60\right)\cdot 67 + \left(34 a + 47\right)\cdot 67^{2} + \left(34 a + 47\right)\cdot 67^{3} + \left(20 a + 37\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 a + 9 + \left(2 a + 33\right)\cdot 67 + \left(35 a + 19\right)\cdot 67^{2} + \left(18 a + 6\right)\cdot 67^{3} + \left(4 a + 1\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 8\cdot 67 + 56\cdot 67^{2} + 67^{3} + 11\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(1,2)$ |
$-2$ |
| $9$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$-2$ |
| $4$ |
$3$ |
$(3,4,6)$ |
$1$ |
| $18$ |
$4$ |
$(1,4,2,3)(5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,2,4,5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,2)(3,4,6)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
