Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 42\cdot 71 + 54\cdot 71^{2} + 49\cdot 71^{3} + 62\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 14\cdot 71 + 58\cdot 71^{2} + 24\cdot 71^{3} + 69\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 51\cdot 71 + 15\cdot 71^{2} + 53\cdot 71^{3} + 59\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 21\cdot 71 + 13\cdot 71^{2} + 63\cdot 71^{3} + 68\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 + 49\cdot 71 + 57\cdot 71^{2} + 7\cdot 71^{3} + 2\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 + 19\cdot 71 + 55\cdot 71^{2} + 17\cdot 71^{3} + 11\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 64 + 56\cdot 71 + 12\cdot 71^{2} + 46\cdot 71^{3} + 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 65 + 28\cdot 71 + 16\cdot 71^{2} + 21\cdot 71^{3} + 8\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
