Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 38\cdot 71 + 45\cdot 71^{2} + 52\cdot 71^{3} + 50\cdot 71^{4} + 30\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 27\cdot 71 + 58\cdot 71^{2} + 42\cdot 71^{3} + 46\cdot 71^{4} + 34\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 16\cdot 71 + 17\cdot 71^{2} + 67\cdot 71^{3} + 31\cdot 71^{4} + 68\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 67\cdot 71 + 6\cdot 71^{2} + 2\cdot 71^{3} + 5\cdot 71^{4} + 51\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 3\cdot 71 + 64\cdot 71^{2} + 68\cdot 71^{3} + 65\cdot 71^{4} + 19\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 54\cdot 71 + 53\cdot 71^{2} + 3\cdot 71^{3} + 39\cdot 71^{4} + 2\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 57 + 43\cdot 71 + 12\cdot 71^{2} + 28\cdot 71^{3} + 24\cdot 71^{4} + 36\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 64 + 32\cdot 71 + 25\cdot 71^{2} + 18\cdot 71^{3} + 20\cdot 71^{4} + 40\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,7)(4,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
