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:
| $r_{ 1 }$ |
$=$ |
$ 54 + 17\cdot 71 + 11\cdot 71^{2} + 26\cdot 71^{3} + 23\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 56\cdot 71 + 61\cdot 71^{2} + 21\cdot 71^{3} + 44\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 65\cdot 71 + 39\cdot 71^{2} + 37\cdot 71^{3} + 35\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 1 + \left(41 a + 45\right)\cdot 71 + \left(25 a + 3\right)\cdot 71^{2} + \left(3 a + 39\right)\cdot 71^{3} + \left(26 a + 47\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 44\cdot 71 + 11\cdot 71^{2} + 68\cdot 71^{3} + 36\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 70 a + 3 + \left(29 a + 55\right)\cdot 71 + \left(45 a + 13\right)\cdot 71^{2} + \left(67 a + 20\right)\cdot 71^{3} + \left(44 a + 25\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$10$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $15$ |
$2$ |
$(1,2)$ |
$2$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
