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:
| $r_{ 1 }$ |
$=$ |
$ 28 a + 45 + \left(27 a + 49\right)\cdot 71 + \left(6 a + 17\right)\cdot 71^{2} + \left(2 a + 56\right)\cdot 71^{3} + \left(6 a + 68\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a + 21 + \left(14 a + 38\right)\cdot 71 + \left(2 a + 53\right)\cdot 71^{2} + \left(20 a + 22\right)\cdot 71^{3} + \left(68 a + 57\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 a + 56 + \left(59 a + 22\right)\cdot 71 + 8 a\cdot 71^{2} + \left(19 a + 34\right)\cdot 71^{3} + \left(14 a + 27\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 60 + \left(56 a + 12\right)\cdot 71 + \left(68 a + 43\right)\cdot 71^{2} + \left(50 a + 60\right)\cdot 71^{3} + \left(2 a + 31\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 a + 30 + \left(43 a + 5\right)\cdot 71 + \left(64 a + 3\right)\cdot 71^{2} + \left(68 a + 54\right)\cdot 71^{3} + \left(64 a + 7\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 8 + \left(11 a + 24\right)\cdot 71 + \left(62 a + 29\right)\cdot 71^{2} + \left(51 a + 63\right)\cdot 71^{3} + \left(56 a + 36\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 59\cdot 71 + 65\cdot 71^{2} + 63\cdot 71^{3} + 53\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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$6$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$2$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$2$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$-1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
