Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 347 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 27\cdot 347 + 187\cdot 347^{2} + 57\cdot 347^{3} + 183\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 131 + 215\cdot 347 + 18\cdot 347^{2} + 339\cdot 347^{3} + 234\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 178 + 36\cdot 347 + 130\cdot 347^{2} + 188\cdot 347^{3} + 279\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 190 + 168\cdot 347 + 193\cdot 347^{2} + 36\cdot 347^{3} + 18\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 191 + 246\cdot 347 + 164\cdot 347^{2} + 72\cdot 347^{3} + 325\cdot 347^{4} +O\left(347^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
