Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 18\cdot 71 + 21\cdot 71^{2} + 28\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 44\cdot 71 + 45\cdot 71^{2} + 69\cdot 71^{3} + 37\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 28\cdot 71 + 13\cdot 71^{2} + 14\cdot 71^{3} + 3\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 + 50\cdot 71 + 61\cdot 71^{2} + 57\cdot 71^{3} + 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $4$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
