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 + 58\cdot 71 + 14\cdot 71^{2} + 19\cdot 71^{3} + 12\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 52 + \left(48 a + 4\right)\cdot 71 + \left(32 a + 13\right)\cdot 71^{2} + \left(48 a + 17\right)\cdot 71^{3} + \left(60 a + 68\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 41 + \left(22 a + 68\right)\cdot 71 + \left(54 a + 55\right)\cdot 71^{2} + \left(15 a + 1\right)\cdot 71^{3} + \left(10 a + 27\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 a + 24 + \left(48 a + 15\right)\cdot 71 + 16 a\cdot 71^{2} + \left(55 a + 50\right)\cdot 71^{3} + \left(60 a + 31\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 48\cdot 71 + 27\cdot 71^{2} + 43\cdot 71^{3} + 3\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 a + 5 + \left(22 a + 18\right)\cdot 71 + \left(38 a + 30\right)\cdot 71^{2} + \left(22 a + 10\right)\cdot 71^{3} + \left(10 a + 70\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)$ |
| $(1,2)(3,5)(4,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $6$ |
$2$ |
$(3,4)$ |
$-2$ |
| $9$ |
$2$ |
$(3,4)(5,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,4)$ |
$1$ |
| $4$ |
$3$ |
$(1,3,4)(2,5,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,2)(3,6,4,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,5,3,6,4,2)$ |
$0$ |
| $12$ |
$6$ |
$(2,5,6)(3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
