Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 a + 45 + \left(55 a + 3\right)\cdot 67 + \left(12 a + 2\right)\cdot 67^{2} + \left(65 a + 10\right)\cdot 67^{3} + \left(30 a + 4\right)\cdot 67^{4} + \left(30 a + 55\right)\cdot 67^{5} + \left(9 a + 29\right)\cdot 67^{6} + 2 a\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 14 + \left(48 a + 9\right)\cdot 67 + \left(52 a + 19\right)\cdot 67^{2} + \left(45 a + 35\right)\cdot 67^{3} + \left(18 a + 52\right)\cdot 67^{4} + \left(31 a + 13\right)\cdot 67^{5} + \left(41 a + 33\right)\cdot 67^{6} + \left(7 a + 5\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a + 23 + \left(11 a + 63\right)\cdot 67 + \left(54 a + 64\right)\cdot 67^{2} + \left(a + 56\right)\cdot 67^{3} + \left(36 a + 62\right)\cdot 67^{4} + \left(36 a + 11\right)\cdot 67^{5} + \left(57 a + 37\right)\cdot 67^{6} + \left(64 a + 66\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a + 54 + \left(18 a + 57\right)\cdot 67 + \left(14 a + 47\right)\cdot 67^{2} + \left(21 a + 31\right)\cdot 67^{3} + \left(48 a + 14\right)\cdot 67^{4} + \left(35 a + 53\right)\cdot 67^{5} + \left(25 a + 33\right)\cdot 67^{6} + \left(59 a + 61\right)\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 32\cdot 67 + 34\cdot 67^{2} + 10\cdot 67^{3} + 40\cdot 67^{4} + 50\cdot 67^{5} + 43\cdot 67^{6} + 48\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 34\cdot 67 + 32\cdot 67^{2} + 56\cdot 67^{3} + 26\cdot 67^{4} + 16\cdot 67^{5} + 23\cdot 67^{6} + 18\cdot 67^{7} +O\left(67^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2,5)(3,4,6)$ |
| $(2,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,3)(2,4)$ |
$-1$ |
| $6$ |
$2$ |
$(2,5)(4,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(2,5)(4,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,4,3,2)$ |
$1$ |
| $6$ |
$4$ |
$(1,6,3,5)(2,4)$ |
$-1$ |
| $8$ |
$6$ |
$(1,4,6,3,2,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
