Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 35\cdot 71 + 22\cdot 71^{2} + 64\cdot 71^{3} + 68\cdot 71^{4} + 71^{5} + 69\cdot 71^{6} + 27\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 60\cdot 71 + 71^{2} + 29\cdot 71^{3} + 48\cdot 71^{4} + 41\cdot 71^{5} + 58\cdot 71^{6} + 16\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 4\cdot 71 + 70\cdot 71^{2} + 38\cdot 71^{3} + 62\cdot 71^{4} + 40\cdot 71^{5} + 18\cdot 71^{6} + 60\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 67\cdot 71 + 64\cdot 71^{2} + 37\cdot 71^{3} + 50\cdot 71^{4} + 6\cdot 71^{5} + 49\cdot 71^{6} + 24\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 3\cdot 71 + 6\cdot 71^{2} + 33\cdot 71^{3} + 20\cdot 71^{4} + 64\cdot 71^{5} + 21\cdot 71^{6} + 46\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 66\cdot 71 + 32\cdot 71^{3} + 8\cdot 71^{4} + 30\cdot 71^{5} + 52\cdot 71^{6} + 10\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 10\cdot 71 + 69\cdot 71^{2} + 41\cdot 71^{3} + 22\cdot 71^{4} + 29\cdot 71^{5} + 12\cdot 71^{6} + 54\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 + 35\cdot 71 + 48\cdot 71^{2} + 6\cdot 71^{3} + 2\cdot 71^{4} + 69\cdot 71^{5} + 71^{6} + 43\cdot 71^{7} +O\left(71^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,7)(2,8)(3,6)$ |
| $(1,7,8,2)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(3,6)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
