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:
| $r_{ 1 }$ |
$=$ |
$ 45 a + 63 + \left(45 a + 37\right)\cdot 73 + \left(31 a + 62\right)\cdot 73^{2} + 13\cdot 73^{3} + \left(6 a + 31\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 40\cdot 73 + 12\cdot 73^{2} + 64\cdot 73^{3} + 9\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 59 a + 66 + \left(55 a + 56\right)\cdot 73 + \left(3 a + 72\right)\cdot 73^{2} + 18\cdot 73^{3} + \left(65 a + 53\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 52 + \left(27 a + 56\right)\cdot 73 + \left(41 a + 38\right)\cdot 73^{2} + \left(72 a + 56\right)\cdot 73^{3} + \left(66 a + 48\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a + 24 + \left(17 a + 19\right)\cdot 73 + \left(69 a + 28\right)\cdot 73^{2} + \left(72 a + 15\right)\cdot 73^{3} + \left(7 a + 29\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 7\cdot 73 + 4\cdot 73^{2} + 50\cdot 73^{3} + 46\cdot 73^{4} +O\left(73^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,4,6,2,5,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,3)(4,6)$ | $1$ |
| $20$ | $3$ | $(1,6,5)(2,3,4)$ | $-1$ |
| $30$ | $4$ | $(1,4,3,6)$ | $1$ |
| $24$ | $5$ | $(1,5,4,6,2)$ | $0$ |
| $20$ | $6$ | $(1,4,6,2,5,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
