Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 64\cdot 71 + 69\cdot 71^{2} + 50\cdot 71^{3} + 52\cdot 71^{4} + 26\cdot 71^{5} + 29\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 53\cdot 71 + 48\cdot 71^{2} + 51\cdot 71^{3} + 56\cdot 71^{4} + 37\cdot 71^{5} + 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 17\cdot 71 + 22\cdot 71^{2} + 19\cdot 71^{3} + 14\cdot 71^{4} + 33\cdot 71^{5} + 69\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 6\cdot 71 + 71^{2} + 20\cdot 71^{3} + 18\cdot 71^{4} + 44\cdot 71^{5} + 41\cdot 71^{6} +O\left(71^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
