Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 44 + 45\cdot 73 + 72\cdot 73^{2} + 27\cdot 73^{3} + 32\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 + 10\cdot 73 + 61\cdot 73^{2} + 42\cdot 73^{3} + 43\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 + 60\cdot 73 + 39\cdot 73^{2} + 39\cdot 73^{3} + 33\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 51\cdot 73 + 40\cdot 73^{2} + 13\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 16\cdot 73 + 29\cdot 73^{2} + 28\cdot 73^{3} + 26\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 + 16\cdot 73 + 36\cdot 73^{2} + 64\cdot 73^{3} + 70\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 61 + 66\cdot 73 + 7\cdot 73^{2} + 25\cdot 73^{3} + 16\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 71 + 22\cdot 73 + 4\cdot 73^{2} + 50\cdot 73^{3} + 53\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,6)(4,7,5,8)$ |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,4)(2,5)(3,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,5)(3,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)(3,5)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,6)(4,7,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
