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:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 33 + \left(45 a + 57\right)\cdot 73 + \left(35 a + 18\right)\cdot 73^{2} + \left(26 a + 5\right)\cdot 73^{3} + \left(36 a + 63\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 28 + \left(27 a + 22\right)\cdot 73 + \left(37 a + 24\right)\cdot 73^{2} + \left(46 a + 6\right)\cdot 73^{3} + \left(36 a + 14\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 a + 33 + \left(27 a + 42\right)\cdot 73 + \left(37 a + 60\right)\cdot 73^{2} + \left(46 a + 46\right)\cdot 73^{3} + 36 a\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 65 + \left(45 a + 16\right)\cdot 73 + \left(35 a + 72\right)\cdot 73^{2} + \left(26 a + 2\right)\cdot 73^{3} + \left(36 a + 64\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 60 + \left(45 a + 69\right)\cdot 73 + \left(35 a + 35\right)\cdot 73^{2} + \left(26 a + 35\right)\cdot 73^{3} + \left(36 a + 4\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 a + 1 + \left(27 a + 10\right)\cdot 73 + \left(37 a + 7\right)\cdot 73^{2} + \left(46 a + 49\right)\cdot 73^{3} + \left(36 a + 72\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,5)(3,4)$ |
| $(1,2,4,6,5,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,4,5)(2,6,3)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,5,4)(2,3,6)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,2,4,6,5,3)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $1$ |
$6$ |
$(1,3,5,6,4,2)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
