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:
| $r_{ 1 }$ |
$=$ |
$ 50 a + 21 + \left(15 a + 24\right)\cdot 67 + \left(37 a + 6\right)\cdot 67^{2} + \left(40 a + 17\right)\cdot 67^{3} + \left(40 a + 65\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 a + 42 + \left(63 a + 53\right)\cdot 67 + \left(31 a + 23\right)\cdot 67^{2} + \left(30 a + 4\right)\cdot 67^{3} + \left(47 a + 13\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 59\cdot 67 + 10\cdot 67^{2} + 62\cdot 67^{3} + 31\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 55 + \left(32 a + 2\right)\cdot 67 + \left(51 a + 47\right)\cdot 67^{2} + \left(30 a + 38\right)\cdot 67^{3} + \left(8 a + 31\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 6 + \left(34 a + 28\right)\cdot 67 + \left(15 a + 19\right)\cdot 67^{2} + \left(36 a + 43\right)\cdot 67^{3} + \left(58 a + 34\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 21 + \left(3 a + 62\right)\cdot 67 + \left(35 a + 20\right)\cdot 67^{2} + \left(36 a + 27\right)\cdot 67^{3} + \left(19 a + 38\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 17 a + 20 + \left(51 a + 37\right)\cdot 67 + \left(29 a + 5\right)\cdot 67^{2} + \left(26 a + 8\right)\cdot 67^{3} + \left(26 a + 53\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
