Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 14\cdot 71 + 56\cdot 71^{2} + 44\cdot 71^{3} + 16\cdot 71^{4} + 50\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 36\cdot 71 + 41\cdot 71^{2} + 40\cdot 71^{3} + 32\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 20\cdot 71 + 45\cdot 71^{2} + 20\cdot 71^{3} + 4\cdot 71^{4} + 36\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 70\cdot 71^{2} + 35\cdot 71^{3} + 49\cdot 71^{4} + 23\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 70\cdot 71 + 35\cdot 71^{3} + 21\cdot 71^{4} + 47\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 + 50\cdot 71 + 25\cdot 71^{2} + 50\cdot 71^{3} + 66\cdot 71^{4} + 34\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 62 + 34\cdot 71 + 29\cdot 71^{2} + 30\cdot 71^{3} + 70\cdot 71^{4} + 38\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 63 + 56\cdot 71 + 14\cdot 71^{2} + 26\cdot 71^{3} + 54\cdot 71^{4} + 20\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,6,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,4)(5,8,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
