Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 6\cdot 79 + 42\cdot 79^{2} + 5\cdot 79^{3} + 36\cdot 79^{4} + 76\cdot 79^{5} + 17\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 59\cdot 79 + 71\cdot 79^{2} + 75\cdot 79^{3} + 43\cdot 79^{4} + 73\cdot 79^{5} + 62\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 62\cdot 79 + 31\cdot 79^{2} + 70\cdot 79^{3} + 4\cdot 79^{4} + 76\cdot 79^{5} + 61\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 30\cdot 79 + 12\cdot 79^{2} + 6\cdot 79^{3} + 73\cdot 79^{4} + 10\cdot 79^{5} + 15\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(2,3)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,4,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
