Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 29\cdot 73 + 15\cdot 73^{2} + 35\cdot 73^{3} + 36\cdot 73^{4} + 42\cdot 73^{5} + 23\cdot 73^{6} + 73^{7} + 38\cdot 73^{8} + 48\cdot 73^{9} + 6\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 51\cdot 73 + 36\cdot 73^{2} + 39\cdot 73^{3} + 57\cdot 73^{4} + 11\cdot 73^{5} + 38\cdot 73^{6} + 4\cdot 73^{7} + 39\cdot 73^{8} + 66\cdot 73^{9} + 60\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 67\cdot 73 + 60\cdot 73^{2} + 41\cdot 73^{3} + 2\cdot 73^{4} + 10\cdot 73^{5} + 43\cdot 73^{6} + 26\cdot 73^{7} + 61\cdot 73^{8} + 30\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 71\cdot 73 + 4\cdot 73^{2} + 36\cdot 73^{3} + 15\cdot 73^{4} + 54\cdot 73^{5} + 15\cdot 73^{6} + 42\cdot 73^{7} + 62\cdot 73^{8} + 15\cdot 73^{9} +O\left(73^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 41 + 73 + 68\cdot 73^{2} + 36\cdot 73^{3} + 57\cdot 73^{4} + 18\cdot 73^{5} + 57\cdot 73^{6} + 30\cdot 73^{7} + 10\cdot 73^{8} + 57\cdot 73^{9} + 72\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 5\cdot 73 + 12\cdot 73^{2} + 31\cdot 73^{3} + 70\cdot 73^{4} + 62\cdot 73^{5} + 29\cdot 73^{6} + 46\cdot 73^{7} + 11\cdot 73^{8} + 72\cdot 73^{9} + 42\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 47 + 21\cdot 73 + 36\cdot 73^{2} + 33\cdot 73^{3} + 15\cdot 73^{4} + 61\cdot 73^{5} + 34\cdot 73^{6} + 68\cdot 73^{7} + 33\cdot 73^{8} + 6\cdot 73^{9} + 12\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 61 + 43\cdot 73 + 57\cdot 73^{2} + 37\cdot 73^{3} + 36\cdot 73^{4} + 30\cdot 73^{5} + 49\cdot 73^{6} + 71\cdot 73^{7} + 34\cdot 73^{8} + 24\cdot 73^{9} + 66\cdot 73^{10} +O\left(73^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(2,3)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(2,3)(4,5)(6,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,5,3,8,7,4,6)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,7,5,6,8,2,4,3)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
