Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 79 + 46\cdot 79^{2} + 65\cdot 79^{3} + 67\cdot 79^{4} + 11\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 45\cdot 79 + 59\cdot 79^{2} + 74\cdot 79^{3} + 59\cdot 79^{4} + 10\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 29 + 44\cdot 79 + 67\cdot 79^{2} + 59\cdot 79^{3} + 77\cdot 79^{4} + 37\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 59 + 12\cdot 79 + 59\cdot 79^{2} + 52\cdot 79^{3} + 3\cdot 79^{4} + 58\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 + 11\cdot 79 + 67\cdot 79^{2} + 37\cdot 79^{3} + 21\cdot 79^{4} + 6\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 69 + 20\cdot 79 + 64\cdot 79^{2} + 58\cdot 79^{3} + 8\cdot 79^{4} + 50\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 32\cdot 79 + 8\cdot 79^{2} + 25\cdot 79^{3} + 41\cdot 79^{4} + 11\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 76 + 67\cdot 79 + 22\cdot 79^{2} + 20\cdot 79^{3} + 35\cdot 79^{4} + 50\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,6)(5,7)$ |
| $(1,2,6,5)(3,7,4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,2,6,5)(3,7,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
