Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 421 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 70 + 261\cdot 421 + 276\cdot 421^{2} + 162\cdot 421^{3} + 340\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 247 + 163\cdot 421 + 319\cdot 421^{2} + 234\cdot 421^{3} + 220\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 305 + 12\cdot 421 + 124\cdot 421^{2} + 78\cdot 421^{3} + 343\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 320 + 102\cdot 421 + 21\cdot 421^{2} + 198\cdot 421^{3} + 278\cdot 421^{4} +O\left(421^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 322 + 301\cdot 421 + 100\cdot 421^{2} + 168\cdot 421^{3} + 80\cdot 421^{4} +O\left(421^{ 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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
