Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 59\cdot 71 + 23\cdot 71^{2} + 17\cdot 71^{3} + 44\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 52\cdot 71 + 3\cdot 71^{2} + 34\cdot 71^{3} + 29\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 38\cdot 71 + 50\cdot 71^{2} + 31\cdot 71^{3} + 61\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 31\cdot 71 + 30\cdot 71^{2} + 48\cdot 71^{3} + 46\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 27\cdot 71 + 37\cdot 71^{2} + 38\cdot 71^{3} + 55\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 6\cdot 71 + 64\cdot 71^{2} + 52\cdot 71^{3} + 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 52 + 44\cdot 71 + 23\cdot 71^{2} + 23\cdot 71^{3} + 49\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 69 + 23\cdot 71 + 50\cdot 71^{2} + 37\cdot 71^{3} + 66\cdot 71^{4} +O\left(71^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,7)(3,6,4,8)$ |
| $(1,2)(3,4)(5,7)(6,8)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,5,2,7)(3,6,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
