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:
| $r_{ 1 }$ |
$=$ |
$ 64 a + 64 + \left(41 a + 63\right)\cdot 71 + \left(34 a + 12\right)\cdot 71^{2} + \left(69 a + 51\right)\cdot 71^{3} + \left(67 a + 20\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 22 + \left(a + 63\right)\cdot 71 + \left(45 a + 52\right)\cdot 71^{2} + \left(68 a + 44\right)\cdot 71^{3} + \left(31 a + 29\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a + 50 + \left(29 a + 12\right)\cdot 71 + \left(36 a + 40\right)\cdot 71^{2} + \left(a + 13\right)\cdot 71^{3} + \left(3 a + 16\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + \left(25 a + 11\right)\cdot 71 + \left(58 a + 35\right)\cdot 71^{2} + \left(19 a + 68\right)\cdot 71^{3} + 61 a\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 52 + \left(45 a + 35\right)\cdot 71 + \left(12 a + 55\right)\cdot 71^{2} + \left(51 a + 49\right)\cdot 71^{3} + \left(9 a + 32\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 a + 31 + \left(69 a + 26\right)\cdot 71 + \left(25 a + 70\right)\cdot 71^{2} + \left(2 a + 65\right)\cdot 71^{3} + \left(39 a + 24\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 66 + 70\cdot 71 + 16\cdot 71^{2} + 61\cdot 71^{3} + 16\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $21$ |
$2$ |
$(1,2)$ |
$-4$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$-2$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$-1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
