Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 29\cdot 71 + 47\cdot 71^{2} + 69\cdot 71^{3} + 27\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 51\cdot 71 + 38\cdot 71^{2} + 21\cdot 71^{3} + 67\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 43\cdot 71 + 19\cdot 71^{2} + 40\cdot 71^{3} + 47\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 70\cdot 71 + 14\cdot 71^{2} + 41\cdot 71^{3} + 22\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 2\cdot 71 + 11\cdot 71^{2} + 46\cdot 71^{3} + 48\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 61\cdot 71 + 23\cdot 71^{2} + 32\cdot 71^{3} + 43\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 + 8\cdot 71 + 54\cdot 71^{2} + 49\cdot 71^{3} + 39\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 52 + 17\cdot 71 + 3\cdot 71^{2} + 54\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 |
| $(1,6,8,7)(2,4,5,3)$ |
| $(1,8)(2,5)(3,4)(6,7)$ |
| $(1,4,8,3)(2,7,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $-2$ |
| $2$ | $4$ | $(1,6,8,7)(2,4,5,3)$ | $0$ |
| $2$ | $4$ | $(1,4,8,3)(2,7,5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,5)(3,7,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
