Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 9\cdot 79 + 35\cdot 79^{2} + 7\cdot 79^{3} + 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 57\cdot 79 + 19\cdot 79^{2} + 27\cdot 79^{3} + 25\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 60\cdot 79 + 11\cdot 79^{2} + 6\cdot 79^{3} + 61\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 47\cdot 79 + 66\cdot 79^{2} + 40\cdot 79^{3} + 8\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 + 31\cdot 79 + 12\cdot 79^{2} + 38\cdot 79^{3} + 70\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 + 18\cdot 79 + 67\cdot 79^{2} + 72\cdot 79^{3} + 17\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 72 + 21\cdot 79 + 59\cdot 79^{2} + 51\cdot 79^{3} + 53\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 74 + 69\cdot 79 + 43\cdot 79^{2} + 71\cdot 79^{3} + 77\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,7)(3,8)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,5)(4,8,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
