Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 66\cdot 73 + 3\cdot 73^{2} + 32\cdot 73^{3} + 60\cdot 73^{4} + 65\cdot 73^{5} + 29\cdot 73^{6} + 10\cdot 73^{7} + 24\cdot 73^{8} + 19\cdot 73^{9} + 21\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 48\cdot 73 + 43\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 15\cdot 73^{5} + 51\cdot 73^{6} + 50\cdot 73^{7} + 19\cdot 73^{8} + 30\cdot 73^{9} + 5\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 32\cdot 73 + 36\cdot 73^{2} + 44\cdot 73^{3} + 2\cdot 73^{4} + 26\cdot 73^{5} + 52\cdot 73^{6} + 35\cdot 73^{7} + 53\cdot 73^{8} + 71\cdot 73^{9} +O\left(73^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 31\cdot 73 + 53\cdot 73^{2} + 45\cdot 73^{3} + 33\cdot 73^{4} + 34\cdot 73^{5} + 61\cdot 73^{6} + 44\cdot 73^{7} + 73^{8} + 62\cdot 73^{9} + 27\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 41\cdot 73 + 19\cdot 73^{2} + 27\cdot 73^{3} + 39\cdot 73^{4} + 38\cdot 73^{5} + 11\cdot 73^{6} + 28\cdot 73^{7} + 71\cdot 73^{8} + 10\cdot 73^{9} + 45\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 40\cdot 73 + 36\cdot 73^{2} + 28\cdot 73^{3} + 70\cdot 73^{4} + 46\cdot 73^{5} + 20\cdot 73^{6} + 37\cdot 73^{7} + 19\cdot 73^{8} + 73^{9} + 72\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 62 + 24\cdot 73 + 29\cdot 73^{2} + 27\cdot 73^{3} + 26\cdot 73^{4} + 57\cdot 73^{5} + 21\cdot 73^{6} + 22\cdot 73^{7} + 53\cdot 73^{8} + 42\cdot 73^{9} + 67\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 70 + 6\cdot 73 + 69\cdot 73^{2} + 40\cdot 73^{3} + 12\cdot 73^{4} + 7\cdot 73^{5} + 43\cdot 73^{6} + 62\cdot 73^{7} + 48\cdot 73^{8} + 53\cdot 73^{9} + 51\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,7,8,2)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
