Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 39 a + 28 + \left(49 a + 25\right)\cdot 73 + \left(2 a + 71\right)\cdot 73^{2} + \left(48 a + 20\right)\cdot 73^{3} + \left(55 a + 50\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 28 + \left(67 a + 43\right)\cdot 73 + 37\cdot 73^{2} + \left(38 a + 30\right)\cdot 73^{3} + \left(49 a + 25\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 a + 72 + \left(23 a + 61\right)\cdot 73 + \left(70 a + 29\right)\cdot 73^{2} + \left(24 a + 16\right)\cdot 73^{3} + \left(17 a + 23\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 6\cdot 73 + 42\cdot 73^{2} + 36\cdot 73^{3} + 34\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 2 + \left(5 a + 60\right)\cdot 73 + \left(72 a + 45\right)\cdot 73^{2} + \left(34 a + 70\right)\cdot 73^{3} + \left(23 a + 62\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 19 + 21\cdot 73 + 65\cdot 73^{2} + 43\cdot 73^{3} + 22\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $15$ |
$2$ |
$(1,2)$ |
$1$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
