Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 32\cdot 73 + 55\cdot 73^{2} + 61\cdot 73^{3} + 29\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 58\cdot 73 + 44\cdot 73^{3} + 46\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 41\cdot 73 + 3\cdot 73^{2} + 16\cdot 73^{3} + 38\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 50\cdot 73 + 42\cdot 73^{2} + 41\cdot 73^{3} + 59\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 22\cdot 73 + 30\cdot 73^{2} + 31\cdot 73^{3} + 13\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 31\cdot 73 + 69\cdot 73^{2} + 56\cdot 73^{3} + 34\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 14\cdot 73 + 72\cdot 73^{2} + 28\cdot 73^{3} + 26\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 67 + 40\cdot 73 + 17\cdot 73^{2} + 11\cdot 73^{3} + 43\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
