Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 44\cdot 73 + 45\cdot 73^{2} + 6\cdot 73^{3} + 33\cdot 73^{4} + 32\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 9\cdot 73 + 58\cdot 73^{2} + 5\cdot 73^{3} + 13\cdot 73^{4} + 62\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 73 + 20\cdot 73^{2} + 28\cdot 73^{3} + 49\cdot 73^{4} + 37\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 + 39\cdot 73 + 32\cdot 73^{2} + 27\cdot 73^{3} + 29\cdot 73^{4} + 67\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 33\cdot 73 + 40\cdot 73^{2} + 45\cdot 73^{3} + 43\cdot 73^{4} + 5\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 43 + 71\cdot 73 + 52\cdot 73^{2} + 44\cdot 73^{3} + 23\cdot 73^{4} + 35\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 71 + 63\cdot 73 + 14\cdot 73^{2} + 67\cdot 73^{3} + 59\cdot 73^{4} + 10\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 72 + 28\cdot 73 + 27\cdot 73^{2} + 66\cdot 73^{3} + 39\cdot 73^{4} + 40\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,4)(2,3,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
