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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + \left(13 a + 54\right)\cdot 71 + \left(44 a + 46\right)\cdot 71^{2} + \left(42 a + 13\right)\cdot 71^{3} + \left(16 a + 40\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 64 a + 14 + \left(57 a + 2\right)\cdot 71 + \left(26 a + 51\right)\cdot 71^{2} + \left(28 a + 54\right)\cdot 71^{3} + \left(54 a + 30\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 a + 38 + \left(49 a + 29\right)\cdot 71 + \left(52 a + 3\right)\cdot 71^{2} + \left(52 a + 25\right)\cdot 71^{3} + \left(56 a + 47\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a + 63 + \left(45 a + 41\right)\cdot 71 + \left(59 a + 50\right)\cdot 71^{2} + \left(30 a + 62\right)\cdot 71^{3} + \left(38 a + 1\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 + 31\cdot 71 + 19\cdot 71^{2} + 56\cdot 71^{3} + 7\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 a + 58 + \left(25 a + 28\right)\cdot 71 + \left(11 a + 53\right)\cdot 71^{2} + \left(40 a + 64\right)\cdot 71^{3} + \left(32 a + 47\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 a + 31 + \left(21 a + 25\right)\cdot 71 + \left(18 a + 59\right)\cdot 71^{2} + \left(18 a + 6\right)\cdot 71^{3} + \left(14 a + 37\right)\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.
