Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 34\cdot 73 + 14\cdot 73^{2} + 21\cdot 73^{3} + 8\cdot 73^{4} + 18\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 56\cdot 73 + 3\cdot 73^{2} + 40\cdot 73^{3} + 41\cdot 73^{4} + 10\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 10\cdot 73 + 4\cdot 73^{2} + 20\cdot 73^{3} + 13\cdot 73^{4} + 66\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 21\cdot 73 + 59\cdot 73^{2} + 4\cdot 73^{3} + 21\cdot 73^{4} + 57\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 + 51\cdot 73 + 13\cdot 73^{2} + 68\cdot 73^{3} + 51\cdot 73^{4} + 15\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 62\cdot 73 + 68\cdot 73^{2} + 52\cdot 73^{3} + 59\cdot 73^{4} + 6\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 16\cdot 73 + 69\cdot 73^{2} + 32\cdot 73^{3} + 31\cdot 73^{4} + 62\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 71 + 38\cdot 73 + 58\cdot 73^{2} + 51\cdot 73^{3} + 64\cdot 73^{4} + 54\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,7,5,8,6,2,4)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,6,7,4,8,3,2,5)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $-\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
