Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 13\cdot 73 + 29\cdot 73^{2} + 30\cdot 73^{3} + 13\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 + 70\cdot 73 + 13\cdot 73^{2} + 5\cdot 73^{3} + 8\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 37\cdot 73 + 61\cdot 73^{2} + 3\cdot 73^{3} + 21\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 10\cdot 73 + 14\cdot 73^{2} + 49\cdot 73^{3} + 13\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 28\cdot 73 + 56\cdot 73^{2} + 14\cdot 73^{3} + 30\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 25\cdot 73 + 41\cdot 73^{2} + 33\cdot 73^{3} + 30\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 37\cdot 73 + 32\cdot 73^{2} + 53\cdot 73^{3} + 55\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 71 + 69\cdot 73 + 42\cdot 73^{2} + 28\cdot 73^{3} + 46\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,3)(4,7,5,8)$ |
| $(1,4)(2,7)(3,8)(5,6)$ |
| $(1,6)(7,8)$ |
| $(1,6)(2,3)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,7)(3,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $0$ |
| $1$ | $4$ | $(1,7,6,8)(2,5,3,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,6,7)(2,4,3,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,6,3)(4,7,5,8)$ | $0$ |
| $2$ | $4$ | $(1,4,6,5)(2,8,3,7)$ | $0$ |
| $2$ | $4$ | $(1,8,6,7)(2,5,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
