Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 13.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 3\cdot 71 + 49\cdot 71^{2} + 47\cdot 71^{3} + 3\cdot 71^{4} + 7\cdot 71^{5} + 16\cdot 71^{6} + 13\cdot 71^{7} + 54\cdot 71^{8} + 28\cdot 71^{9} + 11\cdot 71^{10} + 10\cdot 71^{11} + 10\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 54\cdot 71 + 3\cdot 71^{2} + 39\cdot 71^{3} + 36\cdot 71^{4} + 22\cdot 71^{5} + 59\cdot 71^{6} + 26\cdot 71^{7} + 36\cdot 71^{8} + 48\cdot 71^{9} + 22\cdot 71^{10} + 15\cdot 71^{11} + 23\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 65\cdot 71 + 56\cdot 71^{2} + 63\cdot 71^{3} + 53\cdot 71^{4} + 9\cdot 71^{5} + 45\cdot 71^{6} + 24\cdot 71^{7} + 3\cdot 71^{8} + 5\cdot 71^{9} + 69\cdot 71^{10} + 15\cdot 71^{11} + 38\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 10\cdot 71 + 42\cdot 71^{2} + 27\cdot 71^{3} + 63\cdot 71^{4} + 27\cdot 71^{5} + 32\cdot 71^{6} + 48\cdot 71^{7} + 2\cdot 71^{8} + 52\cdot 71^{9} + 65\cdot 71^{10} + 39\cdot 71^{11} + 44\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 60\cdot 71 + 28\cdot 71^{2} + 43\cdot 71^{3} + 7\cdot 71^{4} + 43\cdot 71^{5} + 38\cdot 71^{6} + 22\cdot 71^{7} + 68\cdot 71^{8} + 18\cdot 71^{9} + 5\cdot 71^{10} + 31\cdot 71^{11} + 26\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 5\cdot 71 + 14\cdot 71^{2} + 7\cdot 71^{3} + 17\cdot 71^{4} + 61\cdot 71^{5} + 25\cdot 71^{6} + 46\cdot 71^{7} + 67\cdot 71^{8} + 65\cdot 71^{9} + 71^{10} + 55\cdot 71^{11} + 32\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 + 16\cdot 71 + 67\cdot 71^{2} + 31\cdot 71^{3} + 34\cdot 71^{4} + 48\cdot 71^{5} + 11\cdot 71^{6} + 44\cdot 71^{7} + 34\cdot 71^{8} + 22\cdot 71^{9} + 48\cdot 71^{10} + 55\cdot 71^{11} + 47\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 + 67\cdot 71 + 21\cdot 71^{2} + 23\cdot 71^{3} + 67\cdot 71^{4} + 63\cdot 71^{5} + 54\cdot 71^{6} + 57\cdot 71^{7} + 16\cdot 71^{8} + 42\cdot 71^{9} + 59\cdot 71^{10} + 60\cdot 71^{11} + 60\cdot 71^{12} +O\left(71^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,6)(4,5)$ |
| $(2,7)(4,5)$ |
| $(1,7)(2,8)(3,5)(4,6)$ |
| $(1,4,2,3,8,5,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $0$ |
| $4$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
| $4$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
| $4$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
