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:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 46 + \left(25 a + 69\right)\cdot 73 + \left(71 a + 29\right)\cdot 73^{2} + \left(59 a + 26\right)\cdot 73^{3} + \left(46 a + 68\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 53 + \left(64 a + 47\right)\cdot 73 + \left(57 a + 54\right)\cdot 73^{2} + \left(64 a + 29\right)\cdot 73^{3} + \left(19 a + 5\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 a + 51 + \left(47 a + 46\right)\cdot 73 + \left(a + 72\right)\cdot 73^{2} + \left(13 a + 61\right)\cdot 73^{3} + \left(26 a + 2\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 12 + \left(8 a + 60\right)\cdot 73 + \left(15 a + 17\right)\cdot 73^{2} + \left(8 a + 20\right)\cdot 73^{3} + 53 a\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 42\cdot 73 + 65\cdot 73^{2} + 45\cdot 73^{3} + 12\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 25\cdot 73 + 51\cdot 73^{2} + 34\cdot 73^{3} + 56\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(1,3)$ |
$0$ |
| $9$ |
$2$ |
$(1,3)(2,4)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$1$ |
| $4$ |
$3$ |
$(2,4,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,4,3,2)(5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $12$ |
$6$ |
$(1,3)(2,4,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
