Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 419 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 10 + 98\cdot 419 + 364\cdot 419^{2} + 213\cdot 419^{3} + 123\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 69 + 33\cdot 419 + 7\cdot 419^{2} + 239\cdot 419^{3} + 101\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 74 + 101\cdot 419 + 137\cdot 419^{2} + 345\cdot 419^{3} + 194\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 210 + 46\cdot 419 + 391\cdot 419^{2} + 350\cdot 419^{3} + 96\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 281 + 139\cdot 419 + 360\cdot 419^{2} + 274\cdot 419^{3} + 105\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 292 + 35\cdot 419 + 18\cdot 419^{2} + 132\cdot 419^{3} + 122\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 345 + 281\cdot 419 + 92\cdot 419^{2} + 180\cdot 419^{3} + 392\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 398 + 101\cdot 419 + 305\cdot 419^{2} + 358\cdot 419^{3} + 119\cdot 419^{4} +O\left(419^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,6)(3,8)(4,7)$ |
| $(2,6)(4,7)$ |
| $(1,7,5,4)(2,8,6,3)$ |
| $(1,7,3,6)(2,5,4,8)$ |
| $(1,3)(2,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $-4$ |
| $2$ | $2$ | $(1,3)(2,4)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(2,6)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $0$ |
| $4$ | $4$ | $(1,7,5,4)(2,8,6,3)$ | $0$ |
| $4$ | $4$ | $(1,6,3,7)(2,8,4,5)$ | $0$ |
| $4$ | $4$ | $(1,7,3,6)(2,5,4,8)$ | $0$ |
| $4$ | $4$ | $(2,7,6,4)(3,8)$ | $0$ |
| $4$ | $4$ | $(2,4,6,7)(3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
