Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 37\cdot 71 + 66\cdot 71^{2} + 51\cdot 71^{3} + 38\cdot 71^{4} + 64\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 60\cdot 71 + 37\cdot 71^{2} + 59\cdot 71^{3} + 65\cdot 71^{4} + 60\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 35\cdot 71 + 53\cdot 71^{2} + 30\cdot 71^{3} + 62\cdot 71^{4} + 29\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 25\cdot 71 + 15\cdot 71^{2} + 51\cdot 71^{3} + 29\cdot 71^{4} + 48\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 + 45\cdot 71 + 55\cdot 71^{2} + 19\cdot 71^{3} + 41\cdot 71^{4} + 22\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 35\cdot 71 + 17\cdot 71^{2} + 40\cdot 71^{3} + 8\cdot 71^{4} + 41\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 + 10\cdot 71 + 33\cdot 71^{2} + 11\cdot 71^{3} + 5\cdot 71^{4} + 10\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 70 + 33\cdot 71 + 4\cdot 71^{2} + 19\cdot 71^{3} + 32\cdot 71^{4} + 6\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
