Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 48 + 43\cdot 97 + 11\cdot 97^{2} + 59\cdot 97^{3} + 54\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 + 90\cdot 97 + 7\cdot 97^{2} + 50\cdot 97^{3} + 53\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 92 + 59\cdot 97 + 77\cdot 97^{2} + 84\cdot 97^{3} + 85\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
