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 value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,4,6,5,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,5,6,4,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
