Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 239 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 + 157\cdot 239 + 183\cdot 239^{2} + 64\cdot 239^{3} + 132\cdot 239^{4} + 95\cdot 239^{5} + 129\cdot 239^{6} +O\left(239^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 + 235\cdot 239 + 226\cdot 239^{2} + 182\cdot 239^{3} + 232\cdot 239^{4} + 9\cdot 239^{5} + 31\cdot 239^{6} +O\left(239^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 + 233\cdot 239 + 227\cdot 239^{2} + 16\cdot 239^{3} + 81\cdot 239^{4} + 184\cdot 239^{5} + 35\cdot 239^{6} +O\left(239^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 98 + 210\cdot 239 + 114\cdot 239^{2} + 82\cdot 239^{3} + 36\cdot 239^{4} + 55\cdot 239^{5} + 199\cdot 239^{6} +O\left(239^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 141 + 28\cdot 239 + 124\cdot 239^{2} + 156\cdot 239^{3} + 202\cdot 239^{4} + 183\cdot 239^{5} + 39\cdot 239^{6} +O\left(239^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 175 + 5\cdot 239 + 11\cdot 239^{2} + 222\cdot 239^{3} + 157\cdot 239^{4} + 54\cdot 239^{5} + 203\cdot 239^{6} +O\left(239^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 194 + 3\cdot 239 + 12\cdot 239^{2} + 56\cdot 239^{3} + 6\cdot 239^{4} + 229\cdot 239^{5} + 207\cdot 239^{6} +O\left(239^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 216 + 81\cdot 239 + 55\cdot 239^{2} + 174\cdot 239^{3} + 106\cdot 239^{4} + 143\cdot 239^{5} + 109\cdot 239^{6} +O\left(239^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,7,8,6,5,2)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,5,8,4)$ |
| $(1,8)(4,5)$ |
| $(1,5,8,4)(2,3,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,8)(4,5)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,5,8,4)(2,6,7,3)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,4)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,4,8,5)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,8)(2,3,7,6)(4,5)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,8)(2,6,7,3)(4,5)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $4$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,3,4,7,8,6,5,2)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,7,5,3,8,2,4,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
