Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 3\cdot 73 + 68\cdot 73^{2} + 72\cdot 73^{3} + 8\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 41\cdot 73 + 52\cdot 73^{2} + 8\cdot 73^{3} + 47\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 49\cdot 73 + 12\cdot 73^{2} + 3\cdot 73^{3} + 24\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 17\cdot 73 + 10\cdot 73^{2} + 52\cdot 73^{3} + 36\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 41\cdot 73 + 65\cdot 73^{2} + 63\cdot 73^{3} + 52\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 2\cdot 73 + 55\cdot 73^{2} + 17\cdot 73^{3} + 3\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 10\cdot 73 + 15\cdot 73^{2} + 12\cdot 73^{3} + 53\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 71 + 51\cdot 73 + 12\cdot 73^{2} + 61\cdot 73^{3} + 65\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,6)(5,7)$ |
| $(1,2)(3,8)(4,7)(5,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,8)(4,7)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,6,8,7)(2,4,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
