Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 4\cdot 73 + 45\cdot 73^{2} + 4\cdot 73^{3} + 55\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 21\cdot 73 + 47\cdot 73^{2} + 60\cdot 73^{3} + 39\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 42\cdot 73 + 38\cdot 73^{2} + 48\cdot 73^{3} + 23\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 17\cdot 73 + 42\cdot 73^{2} + 52\cdot 73^{3} + 62\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 55\cdot 73 + 30\cdot 73^{2} + 20\cdot 73^{3} + 10\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 30\cdot 73 + 34\cdot 73^{2} + 24\cdot 73^{3} + 49\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 60 + 51\cdot 73 + 25\cdot 73^{2} + 12\cdot 73^{3} + 33\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 67 + 68\cdot 73 + 27\cdot 73^{2} + 68\cdot 73^{3} + 17\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(2,7)(4,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
