Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 18\cdot 71 + 39\cdot 71^{2} + 54\cdot 71^{3} + 67\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 41\cdot 71 + 16\cdot 71^{2} + 8\cdot 71^{3} + 30\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 55\cdot 71 + 67\cdot 71^{2} + 69\cdot 71^{3} + 13\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 7\cdot 71 + 45\cdot 71^{2} + 23\cdot 71^{3} + 47\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 + 45\cdot 71 + 35\cdot 71^{2} + 9\cdot 71^{3} + 27\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 + 64\cdot 71 + 64\cdot 71^{2} + 41\cdot 71^{3} + 2\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 + 52\cdot 71 + 63\cdot 71^{2} + 21\cdot 71^{3} + 24\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 54 + 22\cdot 71^{2} + 54\cdot 71^{3} + 70\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,4,8)(2,7,3,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,5,4,8)(2,7,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
