Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 58\cdot 73 + 52\cdot 73^{2} + 12\cdot 73^{3} + 26\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 70\cdot 73 + 53\cdot 73^{2} + 20\cdot 73^{3} + 48\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 4\cdot 73 + 49\cdot 73^{2} + 43\cdot 73^{3} + 32\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 19\cdot 73^{2} + 35\cdot 73^{3} + 52\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 10\cdot 73 + 25\cdot 73^{2} + 54\cdot 73^{3} + 34\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 13\cdot 73 + 63\cdot 73^{2} + 68\cdot 73^{3} + 38\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 + 51\cdot 73 + 24\cdot 73^{2} + 68\cdot 73^{3} + 68\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 61 + 10\cdot 73 + 4\cdot 73^{2} + 61\cdot 73^{3} + 62\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,5)(6,7)$ |
| $(1,2,3,6)(4,7,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,2,3,6)(4,7,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
