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:
| $r_{ 1 }$ |
$=$ |
$ 66 a + 46 + \left(43 a + 22\right)\cdot 67 + \left(34 a + 36\right)\cdot 67^{2} + \left(54 a + 7\right)\cdot 67^{3} + \left(66 a + 25\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 66 + a\cdot 67 + \left(14 a + 58\right)\cdot 67^{2} + \left(12 a + 39\right)\cdot 67^{3} + \left(27 a + 62\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ a + 42 + \left(23 a + 65\right)\cdot 67 + \left(32 a + 63\right)\cdot 67^{2} + \left(12 a + 56\right)\cdot 67^{3} + 36\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a + 39 + \left(65 a + 62\right)\cdot 67 + \left(52 a + 45\right)\cdot 67^{2} + \left(54 a + 7\right)\cdot 67^{3} + \left(39 a + 25\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 17\cdot 67 + 51\cdot 67^{2} + 60\cdot 67^{3} + 54\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 31\cdot 67 + 12\cdot 67^{2} + 28\cdot 67^{3} + 63\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,3,5)(2,4,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,5)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,6,5,2)$ | $0$ |
| $12$ | $6$ | $(2,4,6)(3,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
