Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 12.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 27\cdot 71 + 17\cdot 71^{2} + 17\cdot 71^{3} + 42\cdot 71^{4} + 26\cdot 71^{5} + 6\cdot 71^{6} + 6\cdot 71^{7} + 66\cdot 71^{8} + 52\cdot 71^{9} + 66\cdot 71^{10} + 11\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 45\cdot 71 + 63\cdot 71^{2} + 29\cdot 71^{3} + 69\cdot 71^{4} + 70\cdot 71^{5} + 2\cdot 71^{6} + 40\cdot 71^{7} + 16\cdot 71^{8} + 53\cdot 71^{9} + 42\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 35\cdot 71 + 36\cdot 71^{2} + 31\cdot 71^{3} + 62\cdot 71^{4} + 43\cdot 71^{5} + 31\cdot 71^{6} + 42\cdot 71^{7} + 25\cdot 71^{8} + 63\cdot 71^{9} + 15\cdot 71^{10} + 18\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 43\cdot 71 + 53\cdot 71^{2} + 28\cdot 71^{3} + 25\cdot 71^{4} + 20\cdot 71^{5} + 69\cdot 71^{6} + 49\cdot 71^{7} + 40\cdot 71^{8} + 45\cdot 71^{9} + 12\cdot 71^{10} + 18\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 27\cdot 71 + 17\cdot 71^{2} + 42\cdot 71^{3} + 45\cdot 71^{4} + 50\cdot 71^{5} + 71^{6} + 21\cdot 71^{7} + 30\cdot 71^{8} + 25\cdot 71^{9} + 58\cdot 71^{10} + 52\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 + 35\cdot 71 + 34\cdot 71^{2} + 39\cdot 71^{3} + 8\cdot 71^{4} + 27\cdot 71^{5} + 39\cdot 71^{6} + 28\cdot 71^{7} + 45\cdot 71^{8} + 7\cdot 71^{9} + 55\cdot 71^{10} + 52\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 + 25\cdot 71 + 7\cdot 71^{2} + 41\cdot 71^{3} + 71^{4} + 68\cdot 71^{6} + 30\cdot 71^{7} + 54\cdot 71^{8} + 17\cdot 71^{9} + 70\cdot 71^{10} + 28\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 + 43\cdot 71 + 53\cdot 71^{2} + 53\cdot 71^{3} + 28\cdot 71^{4} + 44\cdot 71^{5} + 64\cdot 71^{6} + 64\cdot 71^{7} + 4\cdot 71^{8} + 18\cdot 71^{9} + 4\cdot 71^{10} + 59\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,8)(2,5)(4,7)$ |
| $(1,5,3,7,8,4,6,2)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,5)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,4,3,2,8,5,6,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
