Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 28\cdot 71 + 60\cdot 71^{2} + 46\cdot 71^{3} + 20\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 14\cdot 71 + 59\cdot 71^{2} + 2\cdot 71^{3} + 30\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 11\cdot 71 + 4\cdot 71^{2} + 20\cdot 71^{3} + 32\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 + 59\cdot 71 + 69\cdot 71^{2} + 61\cdot 71^{3} + 67\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 67\cdot 71 + 48\cdot 71^{2} + 18\cdot 71^{3} + 67\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 + 41\cdot 71 + 59\cdot 71^{2} + 6\cdot 71^{3} + 34\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 31 + 4\cdot 71 + 19\cdot 71^{2} + 41\cdot 71^{3} + 45\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 38 + 58\cdot 71 + 33\cdot 71^{2} + 14\cdot 71^{3} + 57\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,4)(5,8,6,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,4)(5,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
