Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 521 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 + 195\cdot 521 + 68\cdot 521^{2} + 470\cdot 521^{3} + 368\cdot 521^{4} + 405\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 + 160\cdot 521 + 308\cdot 521^{2} + 401\cdot 521^{3} + 68\cdot 521^{4} + 396\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 + 201\cdot 521 + 507\cdot 521^{2} + 147\cdot 521^{3} + 77\cdot 521^{4} + 424\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 87 + 126\cdot 521 + 130\cdot 521^{2} + 514\cdot 521^{3} + 478\cdot 521^{4} + 61\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 186 + 431\cdot 521 + 421\cdot 521^{2} + 367\cdot 521^{3} + 383\cdot 521^{4} + 347\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 310 + 465\cdot 521 + 313\cdot 521^{2} + 232\cdot 521^{3} + 220\cdot 521^{4} + 300\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 331 + 83\cdot 521 + 390\cdot 521^{2} + 495\cdot 521^{3} + 301\cdot 521^{4} + 164\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 495 + 420\cdot 521 + 464\cdot 521^{2} + 495\cdot 521^{3} + 183\cdot 521^{4} + 504\cdot 521^{5} +O\left(521^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(4,6)(5,7)$ |
| $(1,2,5,4,3,8,7,6)$ |
| $(2,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $-4$ |
| $2$ | $2$ | $(2,8)(4,6)$ | $0$ |
| $4$ | $2$ | $(4,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,3,7)(2,4,8,6)$ | $0$ |
| $2$ | $4$ | $(1,5,3,7)(2,6,8,4)$ | $0$ |
| $4$ | $8$ | $(1,2,5,4,3,8,7,6)$ | $0$ |
| $4$ | $8$ | $(1,4,7,2,3,6,5,8)$ | $0$ |
| $4$ | $8$ | $(1,2,5,6,3,8,7,4)$ | $0$ |
| $4$ | $8$ | $(1,6,7,2,3,4,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
