Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 58\cdot 73 + 35\cdot 73^{2} + 29\cdot 73^{3} + 16\cdot 73^{4} + 30\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 49\cdot 73 + 13\cdot 73^{2} + 72\cdot 73^{3} + 57\cdot 73^{4} + 70\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 7\cdot 73 + 14\cdot 73^{2} + 54\cdot 73^{3} + 6\cdot 73^{4} + 35\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 31\cdot 73 + 37\cdot 73^{2} + 66\cdot 73^{3} + 56\cdot 73^{4} + 43\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 41\cdot 73 + 35\cdot 73^{2} + 6\cdot 73^{3} + 16\cdot 73^{4} + 29\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 65\cdot 73 + 58\cdot 73^{2} + 18\cdot 73^{3} + 66\cdot 73^{4} + 37\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 + 23\cdot 73 + 59\cdot 73^{2} + 15\cdot 73^{4} + 2\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 61 + 14\cdot 73 + 37\cdot 73^{2} + 43\cdot 73^{3} + 56\cdot 73^{4} + 42\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
