Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 69\cdot 73 + 7\cdot 73^{2} + 59\cdot 73^{3} + 16\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 1 + 39\cdot 73 + 4\cdot 73^{2} + 53\cdot 73^{3} + 23\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 + 70\cdot 73 + 57\cdot 73^{2} + 2\cdot 73^{3} + 52\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 3\cdot 73^{2} + 41\cdot 73^{3} + 31\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 + 43\cdot 73 + 72\cdot 73^{2} + 34\cdot 73^{3} + 38\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 + 73 + 53\cdot 73^{2} + 57\cdot 73^{3} + 66\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 + 32\cdot 73 + 12\cdot 73^{2} + 67\cdot 73^{3} + 23\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 60 + 36\cdot 73 + 7\cdot 73^{2} + 49\cdot 73^{3} + 38\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,6)(5,7)$ |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,3)(2,7)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,7)(4,6)(5,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,8)(3,6)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,3,5)(2,4,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
