Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 23 + 64\cdot 79 + 40\cdot 79^{2} + 26\cdot 79^{3} + 2\cdot 79^{4} + 50\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 + 29\cdot 79 + 18\cdot 79^{2} + 73\cdot 79^{3} + 37\cdot 79^{4} + 35\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 49\cdot 79 + 60\cdot 79^{2} + 5\cdot 79^{3} + 41\cdot 79^{4} + 43\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 14\cdot 79 + 38\cdot 79^{2} + 52\cdot 79^{3} + 76\cdot 79^{4} + 28\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
