Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 17\cdot 73 + 41\cdot 73^{2} + 28\cdot 73^{3} + 48\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 + 50\cdot 73 + 64\cdot 73^{2} + 12\cdot 73^{3} + 23\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 + 39\cdot 73 + 11\cdot 73^{2} + 73^{3} + 19\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 65\cdot 73 + 35\cdot 73^{2} + 70\cdot 73^{3} + 62\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 + 71\cdot 73 + 67\cdot 73^{2} + 48\cdot 73^{3} + 54\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 23\cdot 73 + 57\cdot 73^{2} + 45\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 + 54\cdot 73 + 67\cdot 73^{2} + 55\cdot 73^{3} + 55\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 67 + 41\cdot 73 + 18\cdot 73^{2} + 28\cdot 73^{3} + 12\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,8)(3,4)(5,7)$ |
| $(1,5,6,7)(2,3,8,4)$ |
| $(1,2,6,8)(3,5,4,7)$ |
| $(1,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
| $1$ | $4$ | $(1,7,6,5)(2,3,8,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,6,7)(2,4,8,3)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,6,8)(3,5,4,7)$ | $0$ |
| $2$ | $4$ | $(1,5,6,7)(2,3,8,4)$ | $0$ |
| $2$ | $4$ | $(1,3,6,4)(2,7,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
