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:
| $r_{ 1 }$ |
$=$ |
$ 2 + 73 + 33\cdot 73^{2} + 18\cdot 73^{3} + 6\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 a + 24 + \left(67 a + 3\right)\cdot 73 + \left(72 a + 23\right)\cdot 73^{2} + \left(4 a + 67\right)\cdot 73^{3} + \left(9 a + 34\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 72 a + 1 + \left(32 a + 59\right)\cdot 73 + \left(43 a + 7\right)\cdot 73^{2} + \left(29 a + 41\right)\cdot 73^{3} + \left(45 a + 16\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 71 + \left(40 a + 12\right)\cdot 73 + \left(29 a + 32\right)\cdot 73^{2} + \left(43 a + 13\right)\cdot 73^{3} + \left(27 a + 50\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 a + 62 + \left(5 a + 22\right)\cdot 73 + 28\cdot 73^{2} + \left(68 a + 9\right)\cdot 73^{3} + \left(63 a + 57\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 46\cdot 73 + 21\cdot 73^{2} + 69\cdot 73^{3} + 53\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,5)$ |
| $(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(1,3)$ | $2$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $18$ | $4$ | $(1,5,3,2)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(1,3)(2,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
