Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 4\cdot 73 + 30\cdot 73^{2} + 20\cdot 73^{3} + 2\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 18\cdot 73 + 37\cdot 73^{2} + 38\cdot 73^{3} + 14\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 44\cdot 73 + 64\cdot 73^{2} + 17\cdot 73^{3} + 44\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 58\cdot 73 + 71\cdot 73^{2} + 35\cdot 73^{3} + 56\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 + 14\cdot 73 + 73^{2} + 37\cdot 73^{3} + 16\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 28\cdot 73 + 8\cdot 73^{2} + 55\cdot 73^{3} + 28\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 63 + 54\cdot 73 + 35\cdot 73^{2} + 34\cdot 73^{3} + 58\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 68 + 68\cdot 73 + 42\cdot 73^{2} + 52\cdot 73^{3} + 70\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,3)(5,6,8,7)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,4,3)(5,6,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
