Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17\cdot 71 + 70\cdot 71^{2} + 2\cdot 71^{3} + 36\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 1 + 65\cdot 71 + 63\cdot 71^{2} + 12\cdot 71^{3} + 46\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 31\cdot 71 + 20\cdot 71^{2} + 65\cdot 71^{3} + 68\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 42\cdot 71 + 51\cdot 71^{2} + 54\cdot 71^{3} + 59\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 + 9\cdot 71 + 5\cdot 71^{2} + 46\cdot 71^{3} + 8\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 + 3\cdot 71 + 6\cdot 71^{2} + 9\cdot 71^{3} + 38\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 15\cdot 71 + 8\cdot 71^{2} + 32\cdot 71^{3} + 35\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 63 + 28\cdot 71 + 58\cdot 71^{2} + 60\cdot 71^{3} + 61\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,3)(4,5,6,7)$ |
| $(1,4)(2,7)(3,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,6)(3,4)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,3)(4,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
