Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 401 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 275\cdot 401 + 138\cdot 401^{2} + 159\cdot 401^{3} + 214\cdot 401^{4} + 109\cdot 401^{5} +O\left(401^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 87 + 165\cdot 401 + 153\cdot 401^{2} + 249\cdot 401^{3} + 97\cdot 401^{4} + 335\cdot 401^{5} +O\left(401^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 109 + 244\cdot 401 + 182\cdot 401^{2} + 43\cdot 401^{3} + 293\cdot 401^{4} + 28\cdot 401^{5} +O\left(401^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 132 + 185\cdot 401 + 335\cdot 401^{2} + 87\cdot 401^{3} + 78\cdot 401^{4} + 365\cdot 401^{5} +O\left(401^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 271 + 56\cdot 401 + 115\cdot 401^{2} + 292\cdot 401^{3} + 158\cdot 401^{4} + 263\cdot 401^{5} +O\left(401^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 280 + 237\cdot 401 + 87\cdot 401^{2} + 104\cdot 401^{3} + 26\cdot 401^{4} + 168\cdot 401^{5} +O\left(401^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 336 + 335\cdot 401 + 350\cdot 401^{2} + 216\cdot 401^{3} + 252\cdot 401^{4} + 174\cdot 401^{5} +O\left(401^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 372 + 103\cdot 401 + 240\cdot 401^{2} + 49\cdot 401^{3} + 82\cdot 401^{4} + 159\cdot 401^{5} +O\left(401^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,5)(4,6)$ |
| $(1,5,4,7,8,2,6,3)$ |
| $(3,7)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,5)(3,7)(4,6)$ |
$-4$ |
| $2$ |
$2$ |
$(2,5)(3,7)$ |
$0$ |
| $4$ |
$2$ |
$(2,5)(4,6)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,7)(3,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,6)(2,3,5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,6,8,4)(2,3,5,7)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,4,7,8,2,6,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,6,5,8,3,4,2)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,6,7,8,2,4,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,4,5,8,3,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
