Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 36\cdot 61 + 7\cdot 61^{2} + 26\cdot 61^{3} + 33\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 47\cdot 61 + 14\cdot 61^{2} + 3\cdot 61^{3} + 18\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 3\cdot 61 + 60\cdot 61^{2} + 3\cdot 61^{3} + 43\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 + 17\cdot 61 + 12\cdot 61^{2} + 59\cdot 61^{3} + 42\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 + 33\cdot 61 + 37\cdot 61^{2} + 20\cdot 61^{3} + 26\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 52 + 61 + 60\cdot 61^{2} + 24\cdot 61^{3} + 23\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 56 + 28\cdot 61 + 52\cdot 61^{2} + 16\cdot 61^{3} + 38\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 58 + 13\cdot 61 + 60\cdot 61^{2} + 27\cdot 61^{3} + 18\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,5,6,8,2,3,7)$ |
| $(2,4)(6,7)$ |
| $(1,5,8,3)(2,6,4,7)$ |
| $(1,8)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,4)(3,5)(6,7)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,8)(3,5)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,5,8,3)(2,7,4,6)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,3,8,5)(2,6,4,7)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,5,8,3)(2,6,4,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,5,6,8,2,3,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,3,4,8,7,5,2)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,3,6,8,4,5,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,5,2,8,7,3,4)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
