Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 13.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 51\cdot 71 + 10\cdot 71^{2} + 43\cdot 71^{3} + 50\cdot 71^{4} + 22\cdot 71^{5} + 37\cdot 71^{6} + 9\cdot 71^{7} + 36\cdot 71^{8} + 56\cdot 71^{9} + 24\cdot 71^{10} + 13\cdot 71^{11} + 13\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 31\cdot 71 + 57\cdot 71^{2} + 3\cdot 71^{3} + 21\cdot 71^{4} + 11\cdot 71^{5} + 57\cdot 71^{6} + 10\cdot 71^{7} + 57\cdot 71^{8} + 69\cdot 71^{9} + 63\cdot 71^{10} + 56\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 19\cdot 71 + 68\cdot 71^{2} + 56\cdot 71^{3} + 42\cdot 71^{4} + 56\cdot 71^{5} + 14\cdot 71^{6} + 7\cdot 71^{7} + 53\cdot 71^{8} + 15\cdot 71^{9} + 67\cdot 71^{10} + 23\cdot 71^{11} + 25\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 50\cdot 71 + 27\cdot 71^{2} + 4\cdot 71^{3} + 9\cdot 71^{4} + 56\cdot 71^{5} + 45\cdot 71^{6} + 33\cdot 71^{7} + 47\cdot 71^{8} + 49\cdot 71^{9} + 58\cdot 71^{10} + 5\cdot 71^{11} + 26\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 + 20\cdot 71 + 43\cdot 71^{2} + 66\cdot 71^{3} + 61\cdot 71^{4} + 14\cdot 71^{5} + 25\cdot 71^{6} + 37\cdot 71^{7} + 23\cdot 71^{8} + 21\cdot 71^{9} + 12\cdot 71^{10} + 65\cdot 71^{11} + 44\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 51\cdot 71 + 2\cdot 71^{2} + 14\cdot 71^{3} + 28\cdot 71^{4} + 14\cdot 71^{5} + 56\cdot 71^{6} + 63\cdot 71^{7} + 17\cdot 71^{8} + 55\cdot 71^{9} + 3\cdot 71^{10} + 47\cdot 71^{11} + 45\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 39\cdot 71 + 13\cdot 71^{2} + 67\cdot 71^{3} + 49\cdot 71^{4} + 59\cdot 71^{5} + 13\cdot 71^{6} + 60\cdot 71^{7} + 13\cdot 71^{8} + 71^{9} + 7\cdot 71^{10} + 70\cdot 71^{11} + 14\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 19\cdot 71 + 60\cdot 71^{2} + 27\cdot 71^{3} + 20\cdot 71^{4} + 48\cdot 71^{5} + 33\cdot 71^{6} + 61\cdot 71^{7} + 34\cdot 71^{8} + 14\cdot 71^{9} + 46\cdot 71^{10} + 57\cdot 71^{11} + 57\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,8,3)(2,4,7,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,8,3)(2,4,7,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
