Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 14.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 5\cdot 73 + 47\cdot 73^{2} + 20\cdot 73^{3} + 5\cdot 73^{4} + 12\cdot 73^{5} + 32\cdot 73^{6} + 66\cdot 73^{7} + 67\cdot 73^{8} + 61\cdot 73^{9} + 58\cdot 73^{10} + 66\cdot 73^{11} + 73^{12} + 57\cdot 73^{13} +O\left(73^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 31\cdot 73 + 40\cdot 73^{2} + 27\cdot 73^{3} + 45\cdot 73^{4} + 32\cdot 73^{5} + 56\cdot 73^{6} + 55\cdot 73^{7} + 44\cdot 73^{8} + 23\cdot 73^{9} + 67\cdot 73^{10} + 21\cdot 73^{11} + 13\cdot 73^{12} + 55\cdot 73^{13} +O\left(73^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 48\cdot 73 + 60\cdot 73^{2} + 49\cdot 73^{3} + 34\cdot 73^{4} + 41\cdot 73^{5} + 52\cdot 73^{6} + 72\cdot 73^{7} + 35\cdot 73^{8} + 53\cdot 73^{9} + 62\cdot 73^{10} + 23\cdot 73^{11} + 42\cdot 73^{12} + 35\cdot 73^{13} +O\left(73^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 21\cdot 73 + 38\cdot 73^{2} + 11\cdot 73^{3} + 31\cdot 73^{4} + 49\cdot 73^{5} + 38\cdot 73^{6} + 3\cdot 73^{7} + 37\cdot 73^{8} + 39\cdot 73^{9} + 54\cdot 73^{10} + 43\cdot 73^{11} + 14\cdot 73^{12} + 43\cdot 73^{13} +O\left(73^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 + 51\cdot 73 + 34\cdot 73^{2} + 61\cdot 73^{3} + 41\cdot 73^{4} + 23\cdot 73^{5} + 34\cdot 73^{6} + 69\cdot 73^{7} + 35\cdot 73^{8} + 33\cdot 73^{9} + 18\cdot 73^{10} + 29\cdot 73^{11} + 58\cdot 73^{12} + 29\cdot 73^{13} +O\left(73^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 24\cdot 73 + 12\cdot 73^{2} + 23\cdot 73^{3} + 38\cdot 73^{4} + 31\cdot 73^{5} + 20\cdot 73^{6} + 37\cdot 73^{8} + 19\cdot 73^{9} + 10\cdot 73^{10} + 49\cdot 73^{11} + 30\cdot 73^{12} + 37\cdot 73^{13} +O\left(73^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 71 + 41\cdot 73 + 32\cdot 73^{2} + 45\cdot 73^{3} + 27\cdot 73^{4} + 40\cdot 73^{5} + 16\cdot 73^{6} + 17\cdot 73^{7} + 28\cdot 73^{8} + 49\cdot 73^{9} + 5\cdot 73^{10} + 51\cdot 73^{11} + 59\cdot 73^{12} + 17\cdot 73^{13} +O\left(73^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 72 + 67\cdot 73 + 25\cdot 73^{2} + 52\cdot 73^{3} + 67\cdot 73^{4} + 60\cdot 73^{5} + 40\cdot 73^{6} + 6\cdot 73^{7} + 5\cdot 73^{8} + 11\cdot 73^{9} + 14\cdot 73^{10} + 6\cdot 73^{11} + 71\cdot 73^{12} + 15\cdot 73^{13} +O\left(73^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(5,8)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,4,8,5)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,2,4,3,8,7,5,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
