Jacobian in characteristic 3

[Gabriel Furstenheim Milerud]

I'd like to create an elliptic curve from a degree 3 polynomial without a base point, but when I use the Jacobian method I get a "division by zero" error. 

This is my data:

A=GF(3^2,'c')

S.<a,b,c>=A[]

gS=a^3 - a^2*b + b^3 - a^2*c - a*b*c + b^2*c + a*c^2 + b*c^2 + c^3

Jacobian(gS)

My intuition is that Jacobian tries to compute a short Weierstrass polynomial and this is not necessarily possible in characteristic 3 (I didn't have any problem in characteristic 5). I also checked that the curve was smooth, so it is indeed genus 1

Is there any way to make it work?

This is the error I get:

 

"Traceback (most recent call last):

  File "sag.py", line 16, in <module>

    Jacobian(gS)

  File "/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/jacobian.py", line 116, in Jacobian

    return Jacobian_of_equation(X, **kwds)

  File "/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/jacobian.py", line 225, in Jacobian_of_equation

    f, g = WeierstrassForm(polynomial, variables=variables)

  File "/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/schemes/toric/weierstrass.py", line 506, in WeierstrassForm

    return WeierstrassForm_P2(polynomial, variables)

  File "/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/schemes/toric/weierstrass.py", line 771, in WeierstrassForm_P2

    S = cubic.S_invariant()

  File "/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/rings/invariant_theory.py", line 1769, in S_invariant

    a,b,c,a2,a3,b1,b3,c1,c2,m = self.scaled_coeffs()

  File "/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/rings/invariant_theory.py", line 1753, in scaled_coeffs

    1/F(3)*a[3], 1/F(3)*a[4], 1/F(3)*a[5],

  File "element.pyx", line 1813, in sage.structure.element.RingElement.__div__ (sage/structure/element.c:15219)

  File "coerce.pyx", line 783, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (sage/structure/coerce.c:7325)

  File "element.pyx", line 1811, in sage.structure.element.RingElement.__div__ (sage/structure/element.c:15198)

  File "element_givaro.pyx", line 1201, in sage.rings.finite_rings.element_givaro.FiniteField_givaroElement._div_ (sage/rings/finite_rings/element_givaro.cpp:10695)"

[John Cremona]

It is quite likely that the implementation of Jacobian for plane cubics does not work in characteristic 3 (at least in general).  I had something to do with early implementations but then it was all rewritten in terms of Jacobians, so I cannot remember the details.  Perhaps Volker Braun can comment.  You yourself can look at the code which is run to see what you think -- and of course you are more than welcome to add code for char. 3.

John Cremona

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[Gabriel Furstenheim Milerud]

I've read through the code and this is indeed the problem. It calls WeierstrassForm from schemes.toric which only implements the short version.

Fernando Rodriguez Villegas has some code available for Pari/GP http://www.ma.utexas.edu/users/villegas/cnt/cnt-frames.html in "jacobians" which works in any characteristic.

[Volker Braun]

Indeed I was (am) only interested in the case of char != 2,3. Its should be easy to add the missing cases, tough.