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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 41\cdot 67 + 23\cdot 67^{2} + 62\cdot 67^{3} + 65\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 50 + \left(36 a + 8\right)\cdot 67 + \left(4 a + 44\right)\cdot 67^{2} + \left(3 a + 4\right)\cdot 67^{3} + \left(19 a + 36\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 38 + \left(58 a + 44\right)\cdot 67 + \left(2 a + 7\right)\cdot 67^{2} + \left(47 a + 52\right)\cdot 67^{3} + \left(48 a + 10\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 a + 62 + \left(30 a + 15\right)\cdot 67 + \left(62 a + 26\right)\cdot 67^{2} + \left(63 a + 12\right)\cdot 67^{3} + \left(47 a + 42\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 8 + \left(49 a + 41\right)\cdot 67 + \left(16 a + 60\right)\cdot 67^{2} + \left(40 a + 44\right)\cdot 67^{3} + \left(38 a + 20\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 a + 8 + \left(8 a + 51\right)\cdot 67 + \left(64 a + 27\right)\cdot 67^{2} + \left(19 a + 36\right)\cdot 67^{3} + \left(18 a + 24\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 27 a + 34 + \left(17 a + 65\right)\cdot 67 + \left(50 a + 10\right)\cdot 67^{2} + \left(26 a + 55\right)\cdot 67^{3} + 28 a\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$-4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
