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:
| $r_{ 1 }$ |
$=$ |
$ 22 a + 65 + \left(29 a + 46\right)\cdot 67 + \left(34 a + 63\right)\cdot 67^{2} + \left(56 a + 63\right)\cdot 67^{3} + \left(25 a + 15\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 30 + \left(22 a + 15\right)\cdot 67 + \left(61 a + 30\right)\cdot 67^{2} + \left(6 a + 63\right)\cdot 67^{3} + \left(36 a + 50\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 a + 19 + \left(37 a + 8\right)\cdot 67 + \left(32 a + 38\right)\cdot 67^{2} + \left(10 a + 54\right)\cdot 67^{3} + \left(41 a + 62\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 11\cdot 67 + 32\cdot 67^{2} + 15\cdot 67^{3} + 55\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 51 + \left(44 a + 15\right)\cdot 67 + \left(5 a + 52\right)\cdot 67^{2} + \left(60 a + 29\right)\cdot 67^{3} + \left(30 a + 54\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 35\cdot 67 + 51\cdot 67^{2} + 40\cdot 67^{3} + 28\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,5)$ |
| $(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,3)$ | $2$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $18$ | $4$ | $(1,5,3,2)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(1,3)(2,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
