Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 21\cdot 71 + 39\cdot 71^{2} + 50\cdot 71^{3} + 36\cdot 71^{4} + 70\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 47\cdot 71 + 53\cdot 71^{2} + 70\cdot 71^{3} + 9\cdot 71^{4} + 70\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 62\cdot 71 + 60\cdot 71^{2} + 2\cdot 71^{3} + 48\cdot 71^{4} + 46\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 63\cdot 71 + 3\cdot 71^{2} + 19\cdot 71^{3} + 48\cdot 71^{4} + 6\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 46\cdot 71^{2} + 68\cdot 71^{3} + 64\cdot 71^{4} + 63\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 + 10\cdot 71 + 45\cdot 71^{2} + 71^{3} + 47\cdot 71^{4} + 65\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 43 + 12\cdot 71 + 23\cdot 71^{2} + 24\cdot 71^{3} + 38\cdot 71^{4} + 57\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 46 + 66\cdot 71 + 11\cdot 71^{2} + 46\cdot 71^{3} + 61\cdot 71^{4} + 44\cdot 71^{5} +O\left(71^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,2,3,4,5,6,8)$ |
| $(1,6,4,2)(3,7,8,5)$ |
| $(2,6)(3,5)(7,8)$ |
| $(1,4)(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $-2$ |
| $4$ | $2$ | $(2,6)(3,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,4,6)(3,5,8,7)$ | $0$ |
| $2$ | $8$ | $(1,7,2,3,4,5,6,8)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,6,7,4,8,2,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
