Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 49\cdot 71 + 6\cdot 71^{2} + 5\cdot 71^{3} + 41\cdot 71^{4} + 55\cdot 71^{5} + 48\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 33\cdot 71 + 46\cdot 71^{2} + 71^{3} + 34\cdot 71^{4} + 33\cdot 71^{5} + 19\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 21\cdot 71 + 10\cdot 71^{2} + 9\cdot 71^{3} + 10\cdot 71^{4} + 30\cdot 71^{5} +O\left(71^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 67\cdot 71 + 50\cdot 71^{2} + 58\cdot 71^{3} + 56\cdot 71^{4} + 9\cdot 71^{5} + 63\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 3\cdot 71 + 3\cdot 71^{2} + 69\cdot 71^{3} + 33\cdot 71^{4} + 46\cdot 71^{5} + 29\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 56 + 62\cdot 71 + 37\cdot 71^{2} + 5\cdot 71^{3} + 10\cdot 71^{4} + 43\cdot 71^{5} + 10\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 22\cdot 71 + 59\cdot 71^{2} + 66\cdot 71^{3} + 66\cdot 71^{4} + 26\cdot 71^{5} + 36\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 69 + 22\cdot 71 + 69\cdot 71^{2} + 67\cdot 71^{3} + 30\cdot 71^{4} + 38\cdot 71^{5} + 4\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,5)(2,7,6,8)$ |
| $(1,2)(3,8)(4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,4,5)(2,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
