Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 a + 7 + \left(24 a + 65\right)\cdot 67 + \left(54 a + 58\right)\cdot 67^{2} + \left(17 a + 13\right)\cdot 67^{3} + \left(57 a + 21\right)\cdot 67^{4} + \left(38 a + 64\right)\cdot 67^{5} + \left(47 a + 20\right)\cdot 67^{6} + \left(37 a + 27\right)\cdot 67^{7} + \left(54 a + 50\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 48\cdot 67 + 34\cdot 67^{2} + 42\cdot 67^{3} + 48\cdot 67^{4} + 43\cdot 67^{5} + 17\cdot 67^{6} + 5\cdot 67^{7} + 49\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 58\cdot 67 + 44\cdot 67^{2} + 25\cdot 67^{3} + 10\cdot 67^{4} + 61\cdot 67^{5} + 14\cdot 67^{6} + 3\cdot 67^{7} + 45\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 1 + \left(42 a + 64\right)\cdot 67 + \left(12 a + 50\right)\cdot 67^{2} + \left(49 a + 30\right)\cdot 67^{3} + \left(9 a + 31\right)\cdot 67^{4} + \left(28 a + 28\right)\cdot 67^{5} + \left(19 a + 38\right)\cdot 67^{6} + \left(29 a + 63\right)\cdot 67^{7} + \left(12 a + 29\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 a + 58 + \left(44 a + 14\right)\cdot 67 + \left(47 a + 33\right)\cdot 67^{2} + \left(40 a + 53\right)\cdot 67^{3} + \left(34 a + 62\right)\cdot 67^{4} + \left(53 a + 45\right)\cdot 67^{5} + \left(7 a + 65\right)\cdot 67^{6} + \left(49 a + 56\right)\cdot 67^{7} + \left(60 a + 16\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 25 + \left(22 a + 17\right)\cdot 67 + \left(19 a + 45\right)\cdot 67^{2} + \left(26 a + 34\right)\cdot 67^{3} + \left(32 a + 26\right)\cdot 67^{4} + \left(13 a + 24\right)\cdot 67^{5} + \left(59 a + 43\right)\cdot 67^{6} + \left(17 a + 44\right)\cdot 67^{7} + \left(6 a + 9\right)\cdot 67^{8} +O\left(67^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,2)(3,4,6)$ |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(5,6)$ | $1$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(2,5,3,6)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,5,3,6)$ | $1$ |
| $8$ | $6$ | $(1,5,3,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
