Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 68\cdot 79 + 32\cdot 79^{2} + 40\cdot 79^{3} + 10\cdot 79^{4} + 67\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 74\cdot 79 + 68\cdot 79^{2} + 59\cdot 79^{3} + 4\cdot 79^{4} + 6\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 77\cdot 79 + 16\cdot 79^{2} + 73\cdot 79^{3} + 30\cdot 79^{4} + 26\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 40\cdot 79 + 31\cdot 79^{2} + 36\cdot 79^{3} + 69\cdot 79^{4} + 78\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 + 22\cdot 79 + 51\cdot 79^{2} + 10\cdot 79^{3} + 59\cdot 79^{4} + 19\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 19\cdot 79 + 60\cdot 79^{2} + 66\cdot 79^{3} + 37\cdot 79^{4} + 26\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 71\cdot 79 + 47\cdot 79^{2} + 56\cdot 79^{3} + 78\cdot 79^{4} + 37\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 20\cdot 79 + 6\cdot 79^{2} + 51\cdot 79^{3} + 24\cdot 79^{4} + 53\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,3,7)(2,8,5,4)$ |
| $(1,4,3,8)(2,6,5,7)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
| $(1,8,7,2,3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,4)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,3,6)(2,4,5,8)$ | $0$ |
| $4$ | $4$ | $(1,8,3,4)(2,7,5,6)$ | $0$ |
| $2$ | $8$ | $(1,8,7,2,3,4,6,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,5,3,8,6,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
