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:
| $r_{ 1 }$ |
$=$ |
$ 60 + 3\cdot 73 + 65\cdot 73^{2} + 4\cdot 73^{3} + 64\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 72 a + 45 + \left(67 a + 41\right)\cdot 73 + \left(6 a + 27\right)\cdot 73^{2} + \left(16 a + 13\right)\cdot 73^{3} + \left(11 a + 32\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 50 + \left(55 a + 39\right)\cdot 73 + \left(31 a + 4\right)\cdot 73^{2} + \left(16 a + 67\right)\cdot 73^{3} + \left(22 a + 71\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 a + 58 + \left(17 a + 32\right)\cdot 73 + \left(41 a + 44\right)\cdot 73^{2} + \left(56 a + 11\right)\cdot 73^{3} + \left(50 a + 49\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 24\cdot 73^{2} + 67\cdot 73^{3} + 24\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a + 42 + \left(5 a + 27\right)\cdot 73 + \left(66 a + 53\right)\cdot 73^{2} + \left(56 a + 54\right)\cdot 73^{3} + \left(61 a + 49\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(2,6)$ | $0$ |
| $9$ | $2$ | $(2,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(3,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,3,2,4,6,5)$ | $1$ |
| $12$ | $6$ | $(2,6)(3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
