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:
| $r_{ 1 }$ |
$=$ |
$ 72 a + 38 + \left(65 a + 46\right)\cdot 73 + \left(70 a + 72\right)\cdot 73^{2} + \left(6 a + 24\right)\cdot 73^{3} + \left(23 a + 5\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 23\cdot 73 + 20\cdot 73^{2} + 29\cdot 73^{3} + 21\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a + 45 + \left(46 a + 24\right)\cdot 73 + \left(30 a + 50\right)\cdot 73^{2} + \left(9 a + 37\right)\cdot 73^{3} + \left(70 a + 45\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 35 + \left(7 a + 26\right)\cdot 73 + 2 a\cdot 73^{2} + \left(66 a + 48\right)\cdot 73^{3} + \left(49 a + 67\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 49\cdot 73 + 52\cdot 73^{2} + 43\cdot 73^{3} + 51\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a + 28 + \left(26 a + 48\right)\cdot 73 + \left(42 a + 22\right)\cdot 73^{2} + \left(63 a + 35\right)\cdot 73^{3} + \left(2 a + 27\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6)(2,3,4)$ |
| $(2,5)(3,6)$ |
| $(2,3,5,6)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,5)(3,4)$ | $1$ |
| $8$ | $3$ | $(1,5,6)(2,3,4)$ | $0$ |
| $6$ | $4$ | $(2,3,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
