Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 61 a + 53 + \left(47 a + 59\right)\cdot 67 + \left(61 a + 65\right)\cdot 67^{2} + \left(44 a + 50\right)\cdot 67^{3} + \left(22 a + 21\right)\cdot 67^{4} + \left(16 a + 28\right)\cdot 67^{5} + \left(54 a + 47\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 18 + \left(61 a + 45\right)\cdot 67 + \left(10 a + 28\right)\cdot 67^{2} + \left(7 a + 25\right)\cdot 67^{3} + \left(44 a + 24\right)\cdot 67^{4} + \left(37 a + 14\right)\cdot 67^{5} + \left(40 a + 56\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 54\cdot 67 + 44\cdot 67^{2} + 27\cdot 67^{3} + 39\cdot 67^{4} + 17\cdot 67^{5} + 63\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 63\cdot 67 + 53\cdot 67^{2} + 18\cdot 67^{3} + 55\cdot 67^{4} + 15\cdot 67^{5} + 7\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 24 + \left(5 a + 55\right)\cdot 67 + \left(56 a + 10\right)\cdot 67^{2} + \left(59 a + 43\right)\cdot 67^{3} + \left(22 a + 59\right)\cdot 67^{4} + \left(29 a + 53\right)\cdot 67^{5} + \left(26 a + 46\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 a + 29 + \left(19 a + 56\right)\cdot 67 + \left(5 a + 63\right)\cdot 67^{2} + \left(22 a + 34\right)\cdot 67^{3} + 44 a\cdot 67^{4} + \left(50 a + 4\right)\cdot 67^{5} + \left(12 a + 47\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,4,6,3)$ | $1$ |
| $6$ | $4$ | $(1,4,6,3)(2,5)$ | $-1$ |
| $8$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
