Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 5 }$ to precision 10.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 2\cdot 5^{3} + 5^{4} + 2\cdot 5^{6} + 3\cdot 5^{7} + 3\cdot 5^{8} + 3\cdot 5^{9} +O\left(5^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 2\cdot 5 + 3\cdot 5^{4} + 5^{6} + 2\cdot 5^{7} + 4\cdot 5^{8} + 4\cdot 5^{9} +O\left(5^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 + 2\cdot 5 + 4\cdot 5^{2} + 4\cdot 5^{3} + 5^{4} + 4\cdot 5^{5} + 3\cdot 5^{6} + 2\cdot 5^{7} +O\left(5^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 + 4\cdot 5 + 4\cdot 5^{2} + 2\cdot 5^{3} + 3\cdot 5^{4} + 4\cdot 5^{5} + 2\cdot 5^{6} + 5^{7} + 5^{8} + 5^{9} +O\left(5^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
