Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 45\cdot 73 + 57\cdot 73^{2} + 9\cdot 73^{3} + 66\cdot 73^{4} + 68\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 58\cdot 73 + 51\cdot 73^{2} + 8\cdot 73^{3} + 57\cdot 73^{4} + 27\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 72\cdot 73 + 5\cdot 73^{2} + 10\cdot 73^{3} + 58\cdot 73^{4} + 68\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 46\cdot 73 + 47\cdot 73^{2} + 53\cdot 73^{3} + 20\cdot 73^{4} + 4\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 26\cdot 73 + 25\cdot 73^{2} + 19\cdot 73^{3} + 52\cdot 73^{4} + 68\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 67\cdot 73^{2} + 62\cdot 73^{3} + 14\cdot 73^{4} + 4\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 46 + 14\cdot 73 + 21\cdot 73^{2} + 64\cdot 73^{3} + 15\cdot 73^{4} + 45\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 27\cdot 73 + 15\cdot 73^{2} + 63\cdot 73^{3} + 6\cdot 73^{4} + 4\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,3,7)(2,8,4,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,5)(3,6)(4,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,3,7)(2,8,4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
