Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 521 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 + 307\cdot 521 + 327\cdot 521^{2} + 507\cdot 521^{3} + 434\cdot 521^{4} + 211\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 66 + 395\cdot 521 + 203\cdot 521^{2} + 146\cdot 521^{3} + 40\cdot 521^{4} + 491\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 261 + 518\cdot 521 + 192\cdot 521^{2} + 11\cdot 521^{3} + 130\cdot 521^{4} + 319\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 271 + 244\cdot 521 + 55\cdot 521^{2} + 52\cdot 521^{3} + 320\cdot 521^{4} + 454\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 276 + 336\cdot 521 + 196\cdot 521^{2} + 324\cdot 521^{3} + 453\cdot 521^{4} + 445\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 365 + 398\cdot 521 + 382\cdot 521^{2} + 85\cdot 521^{3} + 316\cdot 521^{4} + 510\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 374 + 284\cdot 521 + 116\cdot 521^{2} + 239\cdot 521^{3} + 264\cdot 521^{4} + 150\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 451 + 119\cdot 521 + 87\cdot 521^{2} + 196\cdot 521^{3} + 124\cdot 521^{4} + 21\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,6)(4,5)$ |
| $(1,6,4,2,7,3,5,8)$ |
| $(2,8)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(2,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(2,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,4,7,5)(2,3,8,6)$ | $0$ |
| $2$ | $4$ | $(1,4,7,5)(2,6,8,3)$ | $0$ |
| $4$ | $8$ | $(1,6,4,2,7,3,5,8)$ | $0$ |
| $4$ | $8$ | $(1,2,5,6,7,8,4,3)$ | $0$ |
| $4$ | $8$ | $(1,6,4,8,7,3,5,2)$ | $0$ |
| $4$ | $8$ | $(1,8,5,6,7,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
