Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 49\cdot 73 + 11\cdot 73^{2} + 10\cdot 73^{3} + 10\cdot 73^{4} + 62\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 7\cdot 73 + 47\cdot 73^{2} + 17\cdot 73^{3} + 39\cdot 73^{4} + 69\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 14\cdot 73 + 66\cdot 73^{2} + 53\cdot 73^{3} + 39\cdot 73^{4} + 3\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 59\cdot 73 + 45\cdot 73^{2} + 14\cdot 73^{3} + 42\cdot 73^{4} + 39\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 13\cdot 73 + 27\cdot 73^{2} + 58\cdot 73^{3} + 30\cdot 73^{4} + 33\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 58\cdot 73 + 6\cdot 73^{2} + 19\cdot 73^{3} + 33\cdot 73^{4} + 69\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 61 + 65\cdot 73 + 25\cdot 73^{2} + 55\cdot 73^{3} + 33\cdot 73^{4} + 3\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 71 + 23\cdot 73 + 61\cdot 73^{2} + 62\cdot 73^{3} + 62\cdot 73^{4} + 10\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
