Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 81\cdot 97 + 53\cdot 97^{2} + 12\cdot 97^{3} + 7\cdot 97^{4} + 10\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 + 67\cdot 97 + 88\cdot 97^{2} + 84\cdot 97^{3} + 63\cdot 97^{4} + 72\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 37\cdot 97 + 21\cdot 97^{2} + 80\cdot 97^{3} + 6\cdot 97^{4} + 59\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 + 64\cdot 97 + 84\cdot 97^{2} + 32\cdot 97^{3} + 51\cdot 97^{4} + 23\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 65 + 31\cdot 97 + 21\cdot 97^{2} + 39\cdot 97^{3} + 95\cdot 97^{4} + 88\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 + 79\cdot 97 + 84\cdot 97^{2} + 20\cdot 97^{3} + 68\cdot 97^{4} + 61\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 85 + 49\cdot 97 + 80\cdot 97^{2} + 35\cdot 97^{3} + 59\cdot 97^{4} + 73\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 96 + 72\cdot 97 + 49\cdot 97^{2} + 81\cdot 97^{3} + 35\cdot 97^{4} + 95\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(4,6)(5,8)$ |
| $(1,6)(2,8)(3,7)(4,5)$ |
| $(1,7,5,8)(2,4,3,6)$ |
| $(1,5)(2,3)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)(7,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,6)(2,8)(3,7)(4,5)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(4,6)(5,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,7,5,8)(2,4,3,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,8,2,5,6,7,3)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,2,7,4,5,3,8,6)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
