Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 61\cdot 73 + 36\cdot 73^{2} + 41\cdot 73^{3} + 43\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 26\cdot 73 + 68\cdot 73^{2} + 66\cdot 73^{3} + 45\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 36\cdot 73 + 33\cdot 73^{2} + 62\cdot 73^{3} + 12\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 73 + 65\cdot 73^{2} + 14\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 50\cdot 73 + 5\cdot 73^{2} + 19\cdot 73^{3} + 55\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 22\cdot 73 + 14\cdot 73^{2} + 24\cdot 73^{3} + 64\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 63 + 60\cdot 73 + 29\cdot 73^{2} + 65\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 71 + 32\cdot 73 + 38\cdot 73^{2} + 70\cdot 73^{3} + 31\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,4,8)(2,7,3,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,5,4,8)(2,7,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
