Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 13.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 43\cdot 73 + 18\cdot 73^{2} + 26\cdot 73^{3} + 42\cdot 73^{4} + 26\cdot 73^{5} + 10\cdot 73^{6} + 19\cdot 73^{7} + 29\cdot 73^{8} + 2\cdot 73^{9} + 45\cdot 73^{10} + 68\cdot 73^{11} + 17\cdot 73^{12} +O\left(73^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 8\cdot 73 + 35\cdot 73^{2} + 43\cdot 73^{3} + 64\cdot 73^{4} + 67\cdot 73^{5} + 37\cdot 73^{6} + 35\cdot 73^{7} + 38\cdot 73^{8} + 19\cdot 73^{9} + 56\cdot 73^{10} + 45\cdot 73^{11} +O\left(73^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 44\cdot 73 + 9\cdot 73^{2} + 57\cdot 73^{3} + 29\cdot 73^{4} + 53\cdot 73^{5} + 56\cdot 73^{6} + 49\cdot 73^{7} + 50\cdot 73^{8} + 8\cdot 73^{9} + 53\cdot 73^{10} + 63\cdot 73^{11} + 45\cdot 73^{12} +O\left(73^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 53\cdot 73 + 69\cdot 73^{2} + 7\cdot 73^{3} + 31\cdot 73^{4} + 24\cdot 73^{5} + 41\cdot 73^{6} + 67\cdot 73^{7} + 6\cdot 73^{8} + 37\cdot 73^{9} + 20\cdot 73^{10} + 48\cdot 73^{11} + 70\cdot 73^{12} +O\left(73^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 19\cdot 73 + 3\cdot 73^{2} + 65\cdot 73^{3} + 41\cdot 73^{4} + 48\cdot 73^{5} + 31\cdot 73^{6} + 5\cdot 73^{7} + 66\cdot 73^{8} + 35\cdot 73^{9} + 52\cdot 73^{10} + 24\cdot 73^{11} + 2\cdot 73^{12} +O\left(73^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 28\cdot 73 + 63\cdot 73^{2} + 15\cdot 73^{3} + 43\cdot 73^{4} + 19\cdot 73^{5} + 16\cdot 73^{6} + 23\cdot 73^{7} + 22\cdot 73^{8} + 64\cdot 73^{9} + 19\cdot 73^{10} + 9\cdot 73^{11} + 27\cdot 73^{12} +O\left(73^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 65 + 64\cdot 73 + 37\cdot 73^{2} + 29\cdot 73^{3} + 8\cdot 73^{4} + 5\cdot 73^{5} + 35\cdot 73^{6} + 37\cdot 73^{7} + 34\cdot 73^{8} + 53\cdot 73^{9} + 16\cdot 73^{10} + 27\cdot 73^{11} + 72\cdot 73^{12} +O\left(73^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 72 + 29\cdot 73 + 54\cdot 73^{2} + 46\cdot 73^{3} + 30\cdot 73^{4} + 46\cdot 73^{5} + 62\cdot 73^{6} + 53\cdot 73^{7} + 43\cdot 73^{8} + 70\cdot 73^{9} + 27\cdot 73^{10} + 4\cdot 73^{11} + 55\cdot 73^{12} +O\left(73^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,3,8,6)(2,4,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
