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:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 33 + \left(27 a + 6\right)\cdot 67 + \left(18 a + 26\right)\cdot 67^{2} + \left(8 a + 30\right)\cdot 67^{3} + \left(25 a + 38\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 7 + \left(39 a + 22\right)\cdot 67 + \left(48 a + 5\right)\cdot 67^{2} + \left(58 a + 45\right)\cdot 67^{3} + \left(41 a + 63\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 65 a + 58 + \left(4 a + 21\right)\cdot 67 + \left(14 a + 33\right)\cdot 67^{2} + \left(18 a + 66\right)\cdot 67^{3} + \left(17 a + 16\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 38\cdot 67 + 35\cdot 67^{2} + 58\cdot 67^{3} + 31\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 50 + \left(62 a + 43\right)\cdot 67 + \left(52 a + 17\right)\cdot 67^{2} + \left(48 a + 58\right)\cdot 67^{3} + 49 a\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 + 67 + 16\cdot 67^{2} + 9\cdot 67^{3} + 49\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,5)(4,6)$ |
| $(3,5,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(3,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,3,2,5,4,6)$ | $-1$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
