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:
| $r_{ 1 }$ |
$=$ |
$ a + 27 + \left(58 a + 64\right)\cdot 67 + \left(49 a + 29\right)\cdot 67^{2} + \left(36 a + 57\right)\cdot 67^{3} + \left(18 a + 15\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 51 + 18\cdot 67 + 34\cdot 67^{2} + 44\cdot 67^{3} + 18\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 66 a + 31 + \left(8 a + 27\right)\cdot 67 + \left(17 a + 37\right)\cdot 67^{2} + \left(30 a + 20\right)\cdot 67^{3} + \left(48 a + 53\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 a + 44 + \left(63 a + 15\right)\cdot 67 + \left(29 a + 29\right)\cdot 67^{2} + \left(43 a + 5\right)\cdot 67^{3} + \left(56 a + 56\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 53\cdot 67 + 51\cdot 67^{2} + 57\cdot 67^{3} + 18\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 a + 35 + \left(3 a + 21\right)\cdot 67 + \left(37 a + 18\right)\cdot 67^{2} + \left(23 a + 15\right)\cdot 67^{3} + \left(10 a + 38\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $2$ |
| $9$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,3)$ | $1$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,4)(2,6,3,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $0$ |
| $12$ | $6$ | $(2,3)(4,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
