Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 4\cdot 73 + 40\cdot 73^{2} + 48\cdot 73^{3} + 44\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 42\cdot 73 + 51\cdot 73^{2} + 49\cdot 73^{3} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 58\cdot 73 + 61\cdot 73^{2} + 7\cdot 73^{3} + 14\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 71\cdot 73 + 60\cdot 73^{2} + 55\cdot 73^{3} + 55\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 12\cdot 73 + 56\cdot 73^{2} + 33\cdot 73^{3} + 31\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 + 41\cdot 73 + 65\cdot 73^{2} + 39\cdot 73^{3} + 13\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 42 + 15\cdot 73 + 37\cdot 73^{2} + 31\cdot 73^{3} + 63\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 59 + 46\cdot 73 + 64\cdot 73^{2} + 24\cdot 73^{3} + 68\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,6)(4,8,5,7)$ |
| $(1,4)(2,7)(3,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,3,6)(4,8,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
