Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 48\cdot 79 + 22\cdot 79^{3} + 43\cdot 79^{4} + 75\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 16\cdot 79 + 67\cdot 79^{2} + 56\cdot 79^{3} + 41\cdot 79^{4} + 6\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 18\cdot 79 + 13\cdot 79^{2} + 54\cdot 79^{3} + 65\cdot 79^{4} + 47\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 9\cdot 79 + 60\cdot 79^{2} + 5\cdot 79^{3} + 64\cdot 79^{4} + 71\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 69\cdot 79 + 18\cdot 79^{2} + 73\cdot 79^{3} + 14\cdot 79^{4} + 7\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 56 + 60\cdot 79 + 65\cdot 79^{2} + 24\cdot 79^{3} + 13\cdot 79^{4} + 31\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 62\cdot 79 + 11\cdot 79^{2} + 22\cdot 79^{3} + 37\cdot 79^{4} + 72\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 30\cdot 79 + 78\cdot 79^{2} + 56\cdot 79^{3} + 35\cdot 79^{4} + 3\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,2,6,5,8,7,3,4)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-1$ |
$-1$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,6,8,3)(2,5,7,4)$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
| $1$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
| $1$ |
$8$ |
$(1,2,6,5,8,7,3,4)$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,5,3,2,8,4,6,7)$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
| $1$ |
$8$ |
$(1,7,6,4,8,2,3,5)$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,4,3,7,8,5,6,2)$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
