Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 71 + 71^{2} + 3\cdot 71^{3} + 60\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 29\cdot 71 + 23\cdot 71^{2} + 5\cdot 71^{3} + 32\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 58\cdot 71 + 7\cdot 71^{2} + 25\cdot 71^{3} + 57\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 53\cdot 71 + 38\cdot 71^{2} + 37\cdot 71^{3} + 63\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 17\cdot 71 + 32\cdot 71^{2} + 33\cdot 71^{3} + 7\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 + 12\cdot 71 + 63\cdot 71^{2} + 45\cdot 71^{3} + 13\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 57 + 41\cdot 71 + 47\cdot 71^{2} + 65\cdot 71^{3} + 38\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 64 + 69\cdot 71 + 69\cdot 71^{2} + 67\cdot 71^{3} + 10\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,4)(5,8,7,6)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,4)(5,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
