CoCalc Public Filessupport / 2015-06-07-022617-genus2reduction.sagews
Authors: Harald Schilly, ℏal Snyder, William A. Stein
Description: Jupyter notebook support/2015-06-04-141749-bokeh.ipynb

︠f45b73b4-6300-4531-aa3d-1de9ccc253f5︠
x = QQ['x'].0
R = genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5)
R.conductor

1416875
genus2reduction?

File: /projects/sage/sage-6.7/src/sage/misc/lazy_import.pyx
Signature : genus2reduction()
Docstring :
Conductor and Reduction Types for Genus 2 Curves.

Use "R = genus2reduction(Q, P)" to obtain reduction information
about the Jacobian of the projective smooth curve defined by y^2 +
Q(x)y = P(x). Type "R?" for further documentation and a description
of how to interpret the local reduction data.

EXAMPLES:

sage: x = QQ['x'].0
sage: R = genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5)
sage: R.conductor
1416875
sage: factor(R.conductor)
5^4 * 2267

This means that only the odd part of the conductor is known.

sage: R.prime_to_2_conductor_only
True

The discriminant is always minimal away from 2, but possibly not at
2.

sage: factor(R.minimal_disc)
2^3 * 5^5 * 2267

Printing R summarizes all the information computed about the curve

sage: R
y^2 + (x^3 - 2*x^2 - 2*x + 1)*y = -5*x^5
A Minimal Equation (away from 2):
y^2 = x^6 - 240*x^4 - 2550*x^3 - 11400*x^2 - 24100*x - 19855
Minimal Discriminant (away from 2):  56675000
Conductor (away from 2): 1416875
Local Data:
p=2
(potential) stable reduction:  (II), j=1
p=5
(potential) stable reduction:  (I)
reduction at p: [V] page 156, (3), f=4
p=2267
(potential) stable reduction:  (II), j=432
reduction at p: [I{1-0-0}] page 170, (1), f=1

Here are some examples of curves with modular Jacobians:

sage: R = genus2reduction(x^3 + x + 1, -2*x^5 - 3*x^2 + 2*x - 2)
sage: factor(R.conductor)
23^2
sage: factor(genus2reduction(x^3 + 1, -x^5 - 3*x^4 + 2*x^2 + 2*x - 2).conductor)
29^2
sage: factor(genus2reduction(x^3 + x + 1, x^5 + 2*x^4 + 2*x^3 + x^2 - x - 1).conductor)
5^6

EXAMPLE:

sage: genus2reduction(0, x^6 + 3*x^3 + 63)
y^2 = x^6 + 3*x^3 + 63
A Minimal Equation (away from 2):
y^2 = x^6 + 3*x^3 + 63
Minimal Discriminant (away from 2):  10628388316852992
Conductor (away from 2): 2893401
Local Data:
p=2
(potential) stable reduction:  (V), j1+j2=0, j1*j2=0
p=3
(potential) stable reduction:  (I)
reduction at p: [III{9}] page 184, (3)^2, f=10
p=7
(potential) stable reduction:  (V), j1+j2=0, j1*j2=0
reduction at p: [I{0}-II-0] page 159, (1), f=2

In the above example, Liu remarks that in fact at p=2, the
reduction is [II-II-0] page 163, (1), f=8. So the conductor of J(C)
is actually 2  * 2893401=5786802.

A MODULAR CURVE:

Consider the modular curve X_1(13) defined by an equation

y^2 + (x^3-x^2-1)y = x^2 - x.

We have:

sage: genus2reduction(x^3-x^2-1, x^2 - x)
y^2 + (x^3 - x^2 - 1)*y = x^2 - x
A Minimal Equation (away from 2):
y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561
Minimal Discriminant (away from 2):  169
Conductor: 169
Local Data:
p=13
(potential) stable reduction:  (V), j1+j2=0, j1*j2=0
reduction at p: [I{0}-II-0] page 159, (1), f=2

So the curve has good reduction at 2. At p=13, the stable reduction
is union of two elliptic curves, and both of them have 0 as modular
invariant. The reduction at 13 is of type [I_0-II-0] (see Namikawa-
Ueno, page 159). It is an elliptic curve with a cusp. The group of
connected components of the Neron model of J(C) is trivial, and the
exponent of the conductor of J(C) at 13 is f=2. The conductor of
J(C) is 13^2. (Note: It is a theorem of Conrad-Edixhoven-Stein that
the component group of J(X_1(p)) is trivial for all primes p.)