> Actually, I wonder if there is some list of the usual commands to use
> on Elliptic Curves. I've been searching for it, but I only found the
> commands to define an Elliptic Curve (just over Q or a finite field,
> but not over a function field, say Q(t) for example) and operate with
> some of points in it. I would like to do other things and I don't
> really know how to. Sorry to bother. Thanks.
>
Hi,
Most of the available commands are methods on an elliptic curve
object. You can use tab completion to see what all is available:
sage: E = EllipticCurve('11a')
sage: E.<tab>
and a huge list of options will pop out. (Here the "<tab>" means hit
tab. This also works in the notebook.) Once you find one you want, you
can use ? to see some documentation, and ?? to see the source:
sage: E.tamagawa_number?
Type: instancemethod
Base Class: <type 'instancemethod'>
String Form: <bound method
EllipticCurve_rational_field.tamagawa_number of Elliptic Curve defined
by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field>
Namespace: Interactive
File:
/sage/local/lib/python2.5/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py
Definition: E.tamagawa_number(self, p)
Docstring:
The Tamagawa number of the elliptic curve at `p`.
This is the order of the component group
`E(QQ_p)/E^0(QQ_p)`.
EXAMPLES::
sage: E = EllipticCurve('11a')
sage: E.tamagawa_number(11)
5
sage: E = EllipticCurve('37b')
sage: E.tamagawa_number(37)
3
sage: E.tamagawa_number??
Type: instancemethod
Base Class: <type 'instancemethod'>
String Form: <bound method
EllipticCurve_rational_field.tamagawa_number of Elliptic Curve defined
by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field>
Namespace: Interactive
File:
/sage/local/lib/python2.5/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py
Definition: E.tamagawa_number(self, p)
Source:
def tamagawa_number(self, p):
r"""
The Tamagawa number of the elliptic curve at `p`.
This is the order of the component group
`E(\QQ_p)/E^0(\QQ_p)`.
EXAMPLES::
sage: E = EllipticCurve('11a')
sage: E.tamagawa_number(11)
5
sage: E = EllipticCurve('37b')
sage: E.tamagawa_number(37)
3
"""
return self.local_data(p).tamagawa_number()
Hope that gets you started ...
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