Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 57\cdot 73 + 16\cdot 73^{2} + 6\cdot 73^{3} + 3\cdot 73^{4} + 10\cdot 73^{5} + 61\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 47\cdot 73 + 29\cdot 73^{2} + 52\cdot 73^{3} + 27\cdot 73^{4} + 36\cdot 73^{5} + 30\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 16\cdot 73 + 43\cdot 73^{2} + 25\cdot 73^{3} + 66\cdot 73^{4} + 8\cdot 73^{5} + 58\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 31\cdot 73 + 24\cdot 73^{2} + 67\cdot 73^{3} + 25\cdot 73^{4} + 2\cdot 73^{5} + 60\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 41\cdot 73 + 48\cdot 73^{2} + 5\cdot 73^{3} + 47\cdot 73^{4} + 70\cdot 73^{5} + 12\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 56\cdot 73 + 29\cdot 73^{2} + 47\cdot 73^{3} + 6\cdot 73^{4} + 64\cdot 73^{5} + 14\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 62 + 25\cdot 73 + 43\cdot 73^{2} + 20\cdot 73^{3} + 45\cdot 73^{4} + 36\cdot 73^{5} + 42\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 68 + 15\cdot 73 + 56\cdot 73^{2} + 66\cdot 73^{3} + 69\cdot 73^{4} + 62\cdot 73^{5} + 11\cdot 73^{6} +O\left(73^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6,8,3)(2,4,7,5)$ |
| $(1,7)(2,8)(4,5)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
