Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4\cdot 73 + 50\cdot 73^{2} + 20\cdot 73^{3} + 57\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 1 + 69\cdot 73 + 22\cdot 73^{2} + 52\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 10\cdot 73 + 18\cdot 73^{2} + 6\cdot 73^{3} + 40\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 57\cdot 73 + 69\cdot 73^{2} + 8\cdot 73^{3} + 56\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 9\cdot 73 + 35\cdot 73^{2} + 5\cdot 73^{3} + 34\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 + 63\cdot 73 + 37\cdot 73^{2} + 67\cdot 73^{3} + 38\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 45 + 15\cdot 73 + 3\cdot 73^{2} + 64\cdot 73^{3} + 16\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 62 + 62\cdot 73 + 54\cdot 73^{2} + 66\cdot 73^{3} + 32\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,7)(5,6)$ |
| $(1,4,8,5)(2,7,3,6)$ |
| $(1,2)(3,8)(4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,7,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
