Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 14.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 13\cdot 71 + 36\cdot 71^{2} + 38\cdot 71^{3} + 38\cdot 71^{4} + 21\cdot 71^{5} + 71^{6} + 37\cdot 71^{7} + 39\cdot 71^{8} + 64\cdot 71^{9} + 51\cdot 71^{10} + 26\cdot 71^{11} + 21\cdot 71^{12} + 25\cdot 71^{13} +O\left(71^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 63\cdot 71 + 63\cdot 71^{2} + 35\cdot 71^{3} + 8\cdot 71^{4} + 5\cdot 71^{5} + 69\cdot 71^{6} + 57\cdot 71^{7} + 31\cdot 71^{8} + 68\cdot 71^{9} + 60\cdot 71^{10} + 31\cdot 71^{11} + 12\cdot 71^{12} + 12\cdot 71^{13} +O\left(71^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 49\cdot 71 + 28\cdot 71^{2} + 32\cdot 71^{3} + 38\cdot 71^{4} + 33\cdot 71^{5} + 49\cdot 71^{6} + 61\cdot 71^{7} + 42\cdot 71^{8} + 19\cdot 71^{9} + 35\cdot 71^{10} + 13\cdot 71^{11} + 22\cdot 71^{12} + 36\cdot 71^{13} +O\left(71^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 + 22\cdot 71 + 36\cdot 71^{2} + 8\cdot 71^{3} + 50\cdot 71^{4} + 8\cdot 71^{5} + 64\cdot 71^{6} + 55\cdot 71^{7} + 44\cdot 71^{8} + 13\cdot 71^{9} + 52\cdot 71^{10} + 18\cdot 71^{11} + 48\cdot 71^{12} + 52\cdot 71^{13} +O\left(71^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 48\cdot 71 + 34\cdot 71^{2} + 62\cdot 71^{3} + 20\cdot 71^{4} + 62\cdot 71^{5} + 6\cdot 71^{6} + 15\cdot 71^{7} + 26\cdot 71^{8} + 57\cdot 71^{9} + 18\cdot 71^{10} + 52\cdot 71^{11} + 22\cdot 71^{12} + 18\cdot 71^{13} +O\left(71^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 60 + 21\cdot 71 + 42\cdot 71^{2} + 38\cdot 71^{3} + 32\cdot 71^{4} + 37\cdot 71^{5} + 21\cdot 71^{6} + 9\cdot 71^{7} + 28\cdot 71^{8} + 51\cdot 71^{9} + 35\cdot 71^{10} + 57\cdot 71^{11} + 48\cdot 71^{12} + 34\cdot 71^{13} +O\left(71^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 64 + 7\cdot 71 + 7\cdot 71^{2} + 35\cdot 71^{3} + 62\cdot 71^{4} + 65\cdot 71^{5} + 71^{6} + 13\cdot 71^{7} + 39\cdot 71^{8} + 2\cdot 71^{9} + 10\cdot 71^{10} + 39\cdot 71^{11} + 58\cdot 71^{12} + 58\cdot 71^{13} +O\left(71^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 65 + 57\cdot 71 + 34\cdot 71^{2} + 32\cdot 71^{3} + 32\cdot 71^{4} + 49\cdot 71^{5} + 69\cdot 71^{6} + 33\cdot 71^{7} + 31\cdot 71^{8} + 6\cdot 71^{9} + 19\cdot 71^{10} + 44\cdot 71^{11} + 49\cdot 71^{12} + 45\cdot 71^{13} +O\left(71^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,7,8,2)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
