Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 69\cdot 73 + 57\cdot 73^{2} + 60\cdot 73^{3} + 58\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 72\cdot 73 + 70\cdot 73^{2} + 37\cdot 73^{3} + 24\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 3\cdot 73 + 72\cdot 73^{2} + 2\cdot 73^{3} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 11\cdot 73 + 46\cdot 73^{2} + 51\cdot 73^{3} + 62\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 + 14\cdot 73 + 59\cdot 73^{2} + 28\cdot 73^{3} + 28\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 + 45\cdot 73 + 41\cdot 73^{2} + 4\cdot 73^{3} + 47\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 54 + 16\cdot 73 + 60\cdot 73^{2} + 51\cdot 73^{3} + 11\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 66 + 58\cdot 73 + 29\cdot 73^{2} + 53\cdot 73^{3} + 58\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,3,5,8)(2,6,4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,3)(4,8)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,5,8)(2,6,4,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
