Opened 11 years ago

# bug in elliptic curve gens() — at Version 7

Reported by: Owned by: rlm cremona major elliptic curves aly.deines, cremona, gagansekhon, weigandt, was, wuthrich, robertwb N/A

[See #15608 for a list of open simon_two_descent tickets]

```sage: a = [1, 0, 1, -1751, -31352]
sage: F = EllipticCurve(a)
sage: FK = EllipticCurve(K, a)
sage: F.gens()
[(52 : 111 : 1)]
sage: FK.gens()
[]
```

This isn't very good, because the default in Sage is proof=True, so one would expect this to be a provable result (until reading the docs of course. But if we try to look harder for the point, we run into a bug with caching:

```sage: FK.gens(lim1=6)
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)

/home/rlmill/<ipython console> in <module>()

/home/rlmill/sage-4.6.2.alpha2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_number_field.pyc in gens(self, verbose, lim1, lim3, limtriv, maxprob, limbigprime)
1772         """
1773
-> 1774         lower,upper,gens = self.simon_two_descent(verbose=verbose,lim1=lim1,lim3=lim3,limtriv=limtriv,maxprob=maxprob,limbigprime=limbigprime)
1775         return gens
1776

/home/rlmill/sage-4.6.2.alpha2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_number_field.pyc in simon_two_descent(self, verbose, lim1, lim3, limtriv, maxprob, limbigprime)
265
266         try:
--> 267             result = self._simon_two_descent_data[lim1,lim3,limtriv,maxprob,limbigprime]
268             if verbose == 0:
269                 return result

KeyError: (6, 50, 10, 20, 30)
```

So two problems: 1) Over Q, if the result is not provable a RuntimeError? is raised. This should be the same here. 2) One can't change parameters due to the way the output is being cached.

### comment:1 Changed 11 years ago by cremona

I'm not sure that, when the gens() function was added over number fields at SD22, we thought it through very well. In particular, I don't think that Simon's code necessarily passes the "proof=True" criteria (but cannot be more specific). Except that the points it returns are points on the curve...

### comment:2 Changed 11 years ago by cremona

The output of simon_two_descent() for EK is

```sage: FK.simon_two_descent()
(1, 1, [])
```

which can be interpreted as follows: he computes the 2-Selmer rank is 1, which gives a valid upper bound for the rank (=1). He fails to find points on 2-coverings, so there are no points returned. *But* he uses the parity conjecture to increase the lower bound from 0 to 1.

So when we decide (in the simon_two_descent()) method) that the output is certain, we need to take this into account.

Secondly, the gens() function for curves over number fields is completely reckless:

```        lower,upper,gens = self.simon_two_descent(verbose=verbose,lim1=lim1,lim3=\
lim3,limtriv=limtriv,maxprob=maxprob,limbigprime=limbigprime)
return gens
```

There is no caching, no checking of Proof, and worst of all, the gens which are returned have not been looked at at all. Just about all you can say about them is that they are points on the curve.

Who let that in? This function needs changing urgently.

### comment:4 Changed 11 years ago by weigandt

If we want to keep a gens() function for elliptic curves over number fields. We're going to need some functionality to check saturation of points over number fields. I'm ccing Robert Bradshaw because he's thought about this.

### comment:5 Changed 8 years ago by wuthrich

It seems that the second problem has disappeared in 6.0. I get now

```sage: FK.gens(lim1=6)
[]
```

That still leaves 1).

Somehow, one should think that it would be best, if the curve remembered that it was defined over a smaller field and that it had found some points of infinite order already. In general, I think it is best if the algorithm would run through all subfield and search for points in there. ... But now I am dreaming.

### comment:6 Changed 8 years ago by wuthrich

I modified the docstring of gens in #13593 . It now says there that the function returns some points of infinite order. In fact Simon's script just gives back what it came across during the various ways to search for points. They are not lin. indep. for instance. And of course there is no guarantee it finds any.

Maybe this ticket should now be rewritten as:

Implement `gens` correctly for elliptic curves over number fields. Extract from the points given by 2-descent (in case it determines the rank) a set of linearly indep. point and then saturate the Mordell-Weil group.

### comment:7 Changed 8 years ago by cremona

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