Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 42\cdot 79 + 27\cdot 79^{2} + 30\cdot 79^{3} + 73\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 63\cdot 79 + 39\cdot 79^{2} + 76\cdot 79^{3} + 39\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 29\cdot 79 + 38\cdot 79^{2} + 5\cdot 79^{3} + 26\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 41\cdot 79 + 21\cdot 79^{2} + 49\cdot 79^{3} + 25\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 + 16\cdot 79 + 55\cdot 79^{2} + 15\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 53 + 79 + 12\cdot 79^{2} + 68\cdot 79^{3} + 15\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 66 + 72\cdot 79 + 42\cdot 79^{2} + 11\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 72 + 48\cdot 79 + 78\cdot 79^{2} + 58\cdot 79^{3} + 40\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,6)(3,4,7,8)$ |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,8,2,4)(3,5,7,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ |
| $2$ | $4$ | $(1,8,2,4)(3,5,7,6)$ | $0$ |
| $2$ | $4$ | $(1,5,2,6)(3,4,7,8)$ | $0$ |
| $2$ | $4$ | $(1,3,2,7)(4,5,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
