Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 40\cdot 71 + 9\cdot 71^{2} + 66\cdot 71^{3} + 17\cdot 71^{4} + 17\cdot 71^{5} + 71^{6} + 66\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 12\cdot 71 + 56\cdot 71^{2} + 42\cdot 71^{3} + 67\cdot 71^{4} + 53\cdot 71^{5} + 51\cdot 71^{6} + 38\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 + 17\cdot 71 + 46\cdot 71^{2} + 29\cdot 71^{3} + 42\cdot 71^{4} + 53\cdot 71^{5} + 61\cdot 71^{6} + 39\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 19\cdot 71 + 28\cdot 71^{2} + 49\cdot 71^{3} + 51\cdot 71^{4} + 50\cdot 71^{5} + 30\cdot 71^{6} + 22\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 + 64\cdot 71 + 48\cdot 71^{2} + 59\cdot 71^{3} + 38\cdot 71^{4} + 32\cdot 71^{5} + 56\cdot 71^{6} + 23\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 19\cdot 71 + 4\cdot 71^{3} + 3\cdot 71^{4} + 26\cdot 71^{5} + 7\cdot 71^{6} + 64\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 + 6\cdot 71 + 19\cdot 71^{2} + 57\cdot 71^{3} + 66\cdot 71^{4} + 57\cdot 71^{5} + 70\cdot 71^{6} + 43\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 47 + 33\cdot 71 + 4\cdot 71^{2} + 46\cdot 71^{3} + 66\cdot 71^{4} + 62\cdot 71^{5} + 3\cdot 71^{6} + 56\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,6,5,3,2,4,8)$ |
| $(4,6)(5,8)$ |
| $(2,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,3)(2,7)(4,6)(5,8)$ |
$-4$ |
| $2$ |
$2$ |
$(2,7)(5,8)$ |
$0$ |
| $4$ |
$2$ |
$(4,6)(5,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(2,8)(3,4)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,3,4)(2,8,7,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,3,4)(2,5,7,8)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,6,5,3,2,4,8)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,4,7,3,8,6,2)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,6,8,3,2,4,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,8,4,7,3,5,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
