Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 31 + 273\cdot 311 + 98\cdot 311^{2} + 6\cdot 311^{3} + 134\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 + 86\cdot 311 + 144\cdot 311^{2} + 42\cdot 311^{3} + 186\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 180 + 92\cdot 311 + 9\cdot 311^{2} + 218\cdot 311^{3} + 208\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 211 + 286\cdot 311 + 240\cdot 311^{2} + 287\cdot 311^{3} + 248\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 238 + 224\cdot 311 + 252\cdot 311^{2} + 82\cdot 311^{3} + 309\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 255 + 108\cdot 311 + 7\cdot 311^{2} + 12\cdot 311^{3} + 75\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 266 + 243\cdot 311 + 128\cdot 311^{2} + 246\cdot 311^{3} + 36\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 309 + 238\cdot 311 + 50\cdot 311^{2} + 37\cdot 311^{3} + 45\cdot 311^{4} +O\left(311^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,4)(3,7)(5,8)$ |
| $(1,7,6,3)(2,8,4,5)$ |
| $(1,2,6,4)(3,8,7,5)$ |
| $(1,5,4,7,6,8,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,7)(5,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,4)(2,6)(5,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,4,6,2)(3,5,7,8)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,3,6,7)(2,5,4,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,4,7,6,8,2,3)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,8,4,3,6,5,2,7)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
