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:
| $r_{ 1 }$ |
$=$ |
$ 55 + 2\cdot 67 + 11\cdot 67^{2} + 36\cdot 67^{3} + 48\cdot 67^{4} + 2\cdot 67^{5} + 5\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 56 + \left(29 a + 27\right)\cdot 67 + \left(23 a + 26\right)\cdot 67^{2} + \left(57 a + 21\right)\cdot 67^{3} + \left(14 a + 62\right)\cdot 67^{4} + \left(34 a + 24\right)\cdot 67^{5} + \left(61 a + 22\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 a + 22 + \left(37 a + 53\right)\cdot 67 + \left(43 a + 23\right)\cdot 67^{2} + \left(9 a + 26\right)\cdot 67^{3} + \left(52 a + 64\right)\cdot 67^{4} + \left(32 a + 12\right)\cdot 67^{5} + \left(5 a + 33\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 62\cdot 67 + 55\cdot 67^{2} + 42\cdot 67^{3} + 42\cdot 67^{4} + 13\cdot 67^{5} + 35\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 54 + \left(65 a + 18\right)\cdot 67 + \left(2 a + 35\right)\cdot 67^{2} + \left(26 a + 53\right)\cdot 67^{3} + 3\cdot 67^{4} + \left(62 a + 50\right)\cdot 67^{5} + \left(34 a + 13\right)\cdot 67^{6} +O\left(67^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 33 + \left(a + 35\right)\cdot 67 + \left(64 a + 48\right)\cdot 67^{2} + \left(40 a + 20\right)\cdot 67^{3} + \left(66 a + 46\right)\cdot 67^{4} + \left(4 a + 29\right)\cdot 67^{5} + \left(32 a + 24\right)\cdot 67^{6} +O\left(67^{ 7 }\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 |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(2,3)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $6$ | $2$ | $(1,5)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,2,4,3)$ | $1$ |
| $6$ | $4$ | $(1,6,4,5)(2,3)$ | $-1$ |
| $8$ | $6$ | $(1,5,2,4,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
