Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 65 + \left(33 a + 33\right)\cdot 67 + \left(42 a + 24\right)\cdot 67^{2} + \left(3 a + 49\right)\cdot 67^{3} + \left(59 a + 32\right)\cdot 67^{4} + \left(64 a + 33\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 a + 30 + \left(33 a + 44\right)\cdot 67 + \left(42 a + 39\right)\cdot 67^{2} + \left(3 a + 45\right)\cdot 67^{3} + \left(59 a + 2\right)\cdot 67^{4} + \left(64 a + 34\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 52\cdot 67 + 17\cdot 67^{2} + 2\cdot 67^{3} + 3\cdot 67^{4} + 30\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a + 38 + \left(33 a + 22\right)\cdot 67 + \left(24 a + 27\right)\cdot 67^{2} + \left(63 a + 21\right)\cdot 67^{3} + \left(7 a + 64\right)\cdot 67^{4} + \left(2 a + 32\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 a + 3 + \left(33 a + 33\right)\cdot 67 + \left(24 a + 42\right)\cdot 67^{2} + \left(63 a + 17\right)\cdot 67^{3} + \left(7 a + 34\right)\cdot 67^{4} + \left(2 a + 33\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 14\cdot 67 + 49\cdot 67^{2} + 64\cdot 67^{3} + 63\cdot 67^{4} + 36\cdot 67^{5} +O\left(67^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(2,3)(4,6)$ |
| $(1,2,3)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,5)$ | $1$ |
| $3$ | $2$ | $(1,5)(2,4)$ | $-1$ |
| $6$ | $2$ | $(2,3)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,3)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,4,5,2)$ | $-1$ |
| $6$ | $4$ | $(1,6,5,3)(2,4)$ | $1$ |
| $8$ | $6$ | $(1,4,6,5,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
