Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 69 a + 69 + \left(67 a + 48\right)\cdot 71 + \left(13 a + 41\right)\cdot 71^{2} + \left(65 a + 56\right)\cdot 71^{3} + \left(32 a + 1\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 54 + 34\cdot 71 + 41\cdot 71^{2} + 48\cdot 71^{3} + 15\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 a + 66 + \left(23 a + 36\right)\cdot 71 + \left(14 a + 45\right)\cdot 71^{2} + \left(22 a + 69\right)\cdot 71^{3} + \left(17 a + 21\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 65 + \left(3 a + 44\right)\cdot 71 + \left(57 a + 1\right)\cdot 71^{2} + \left(5 a + 31\right)\cdot 71^{3} + \left(38 a + 2\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 15\cdot 71 + 56\cdot 71^{2} + 27\cdot 71^{3} + 10\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 60 + \left(47 a + 16\right)\cdot 71 + \left(56 a + 50\right)\cdot 71^{2} + \left(48 a + 28\right)\cdot 71^{3} + \left(53 a + 34\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 29 + 15\cdot 71 + 47\cdot 71^{2} + 21\cdot 71^{3} + 55\cdot 71^{4} +O\left(71^{ 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.
