# Is it error in Sage or mwrank

Sage is unable to compute generators of this elliptic curve:

E=EllipticCurve([0,0,0,-3904/3,-144025/27])
E.gens()


The curve has quite small point (-1373/300, 22927/1000).

Is it error in Sage or in mwrank algorithm?

EDIT: Another example:

E=EllipticCurve([0,0,0,-106/3, -911/27])
E.gens()


It has small point: (-7/3, -6).

Sage is terrible.

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The more diplomatic ,specific, and constructive version of your "sage is terrible" would be: "It looks like there are many opportunities to make "E.gets()" succeed on a lot more elliptic crurves."

It does look like a lot of the higher descent techniques (and 3-descent) that are available in magma, aren't available in sage. It's hard to find someone motivated to do the hard work again -- the first time (for the implementation in magma) the author got a nice paper out of it. That's hard to replicate for the person who'd implement these routines in sage.

( 2020-01-09 03:44:48 +0200 )edit

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The reason why this curve is a little problematic is because it has nontrivial Sha[2]. It would be fairly straightforward to get E.gens() to work: this is a rank 1 elliptic curve and enough of BSD is proved that this can be proven analytically. Sage knows how to do that too:

sage: E.rank(only_use_mwrank=False,proof=True)
1


So the only thing to do is to prove that the point you found isn't divisible by any $n>1$. It is a little surprising that E.gens doesn't seem to have obvious options to help it discover this. The basic elements for implementing this are available.

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@nbruin: I just found even easier case of elliptic curve that Sage is unable to compute its generators. See edit of my question.

( 2020-01-09 01:29:40 +0200 )edit