Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 69\cdot 73^{2} + 61\cdot 73^{3} + 41\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 52\cdot 73 + 15\cdot 73^{2} + 67\cdot 73^{3} + 31\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 55\cdot 73 + 59\cdot 73^{2} + 14\cdot 73^{3} + 70\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 34\cdot 73 + 6\cdot 73^{2} + 20\cdot 73^{3} + 60\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 34\cdot 73 + 4\cdot 73^{2} + 22\cdot 73^{3} + 66\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 50 + 19\cdot 73 + 38\cdot 73^{2} + 30\cdot 73^{3} + 54\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 64 + 18\cdot 73 + 32\cdot 73^{2} + 33\cdot 73^{3} + 62\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 65 + 3\cdot 73 + 66\cdot 73^{2} + 41\cdot 73^{3} + 50\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(5,8)$ |
| $(1,7,4,6)(2,5,3,8)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,2)(3,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $1$ | $4$ | $(1,5,4,8)(2,7,3,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,4,5)(2,6,3,7)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,4,6)(2,5,3,8)$ | $0$ |
| $2$ | $4$ | $(1,2,4,3)(5,7,8,6)$ | $0$ |
| $2$ | $4$ | $(1,8,4,5)(2,7,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
