Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 31\cdot 71 + 9\cdot 71^{2} + 26\cdot 71^{3} + 46\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 44\cdot 71 + 67\cdot 71^{2} + 40\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 53\cdot 71 + 54\cdot 71^{2} + 9\cdot 71^{3} + 57\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 8\cdot 71 + 8\cdot 71^{2} + 40\cdot 71^{3} + 11\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 40\cdot 71 + 2\cdot 71^{2} + 66\cdot 71^{3} + 2\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 46 + 2\cdot 71 + 22\cdot 71^{2} + 8\cdot 71^{3} + 7\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 62\cdot 71 + 61\cdot 71^{2} + 32\cdot 71^{3} + 62\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 56 + 40\cdot 71 + 57\cdot 71^{2} + 28\cdot 71^{3} + 56\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,6)(5,7)$ |
| $(1,2,7,4)(3,6,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,4)(3,5)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,8)(4,6)(5,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,5)(3,4)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,7,4)(3,6,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
