Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 251 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 76\cdot 251 + 117\cdot 251^{2} + 174\cdot 251^{3} + 190\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 + 179\cdot 251 + 225\cdot 251^{2} + 194\cdot 251^{3} + 96\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 71 + 182\cdot 251 + 220\cdot 251^{2} + 135\cdot 251^{3} + 191\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 + 32\cdot 251 + 155\cdot 251^{2} + 230\cdot 251^{3} + 85\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 87 + 127\cdot 251 + 190\cdot 251^{2} + 201\cdot 251^{3} + 25\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 99 + 64\cdot 251 + 189\cdot 251^{2} + 247\cdot 251^{3} + 22\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 163 + 62\cdot 251 + 115\cdot 251^{2} + 209\cdot 251^{3} + 216\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 176 + 28\cdot 251 + 41\cdot 251^{2} + 111\cdot 251^{3} + 173\cdot 251^{4} +O\left(251^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,6)(4,5,7,8)$ |
| $(4,7)(5,8)$ |
| $(3,6)(5,8)$ |
| $(1,7,3,5,2,4,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,7)(5,8)$ |
$-4$ |
| $2$ |
$2$ |
$(4,7)(5,8)$ |
$0$ |
| $4$ |
$2$ |
$(3,6)(5,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,6)(4,5,7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,6)(4,8,7,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,3,5,2,4,6,8)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,6,7,2,8,3,4)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,3,8,2,4,6,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,8,6,7,2,5,3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
