Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 46\cdot 73 + 37\cdot 73^{2} + 3\cdot 73^{3} + 63\cdot 73^{4} + 28\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 50\cdot 73 + 32\cdot 73^{2} + 58\cdot 73^{3} + 4\cdot 73^{4} + 69\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 29\cdot 73 + 54\cdot 73^{2} + 49\cdot 73^{3} + 56\cdot 73^{4} + 70\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 39\cdot 73 + 23\cdot 73^{2} + 41\cdot 73^{3} + 73^{4} + 35\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 33\cdot 73 + 49\cdot 73^{2} + 31\cdot 73^{3} + 71\cdot 73^{4} + 37\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 43\cdot 73 + 18\cdot 73^{2} + 23\cdot 73^{3} + 16\cdot 73^{4} + 2\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 58 + 22\cdot 73 + 40\cdot 73^{2} + 14\cdot 73^{3} + 68\cdot 73^{4} + 3\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 67 + 26\cdot 73 + 35\cdot 73^{2} + 69\cdot 73^{3} + 9\cdot 73^{4} + 44\cdot 73^{5} +O\left(73^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,7,6)(2,3,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
