Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 43 + 189\cdot 311 + 85\cdot 311^{2} + 102\cdot 311^{3} + 254\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 + 213\cdot 311 + 224\cdot 311^{2} + 64\cdot 311^{3} + 205\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 148 + 300\cdot 311 + 57\cdot 311^{2} + 280\cdot 311^{3} + 228\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 150 + 188\cdot 311 + 251\cdot 311^{2} + 311^{3} + 219\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 218 + 26\cdot 311 + 105\cdot 311^{2} + 103\cdot 311^{3} + 270\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 307 + 14\cdot 311 + 208\cdot 311^{2} + 69\cdot 311^{3} + 66\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
