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:
| $r_{ 1 }$ |
$=$ |
$ 48 a + 38 + \left(50 a + 57\right)\cdot 73 + \left(34 a + 9\right)\cdot 73^{2} + \left(27 a + 49\right)\cdot 73^{3} + \left(25 a + 48\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 29 a + 30 + \left(61 a + 68\right)\cdot 73 + \left(23 a + 67\right)\cdot 73^{2} + \left(25 a + 46\right)\cdot 73^{3} + 32 a\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 a + 44 + \left(11 a + 4\right)\cdot 73 + \left(49 a + 5\right)\cdot 73^{2} + \left(47 a + 26\right)\cdot 73^{3} + \left(40 a + 72\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 + 28\cdot 73 + 33\cdot 73^{2} + 52\cdot 73^{3} + 12\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 + 44\cdot 73 + 39\cdot 73^{2} + 20\cdot 73^{3} + 60\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 36 + \left(22 a + 15\right)\cdot 73 + \left(38 a + 63\right)\cdot 73^{2} + \left(45 a + 23\right)\cdot 73^{3} + \left(47 a + 24\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(1,6)(2,4)(3,5)$ |
| $(1,6)(2,3)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,6)(2,3)$ | $-1$ |
| $6$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ |
| $8$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $6$ | $4$ | $(2,5,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
