Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 + 10\cdot 79 + 37\cdot 79^{2} + 23\cdot 79^{3} + 41\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 + 33\cdot 79 + 18\cdot 79^{2} + 9\cdot 79^{3} + 46\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 59 + 19\cdot 79^{2} + 6\cdot 79^{3} + 15\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 70 + 33\cdot 79 + 4\cdot 79^{2} + 40\cdot 79^{3} + 55\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(1,3,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
