Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 69\cdot 97 + 19\cdot 97^{2} + 14\cdot 97^{3} + 70\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 + 24\cdot 97 + 52\cdot 97^{2} + 6\cdot 97^{3} + 13\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 71 + 31\cdot 97 + 71\cdot 97^{2} + 47\cdot 97^{3} + 45\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 83 + 61\cdot 97 + 35\cdot 97^{2} + 58\cdot 97^{3} + 56\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 90 + 6\cdot 97 + 15\cdot 97^{2} + 67\cdot 97^{3} + 8\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
