For the rank $8$ elliptic curve with a-invariants $(0, 0, 1, -23737, 960366)$
sage 5.3 reports analytic rank $4$ in about 2.4 hours.


Almost sure this a bug, so I am interested what other CAS say on the matter of
the analytic rank. Currently testing pari's *ellanalyticrank()* but I have the impression
sage is several times faster than pari on this problem.



> 
> What CASes say about the analytic rank of '457532830151317a1'?
> 
> 
> 


(It might be a good idea to verify my results, I ran them twice).


Session:



```
sage: e= elliptic_curves.rank(8)[0]
sage: e.ainvs()
(0, 0, 1, -23737, 960366)
sage: time e.analytic_rank()
4
Time: CPU 8556.68 s, Wall: 8607.92 s
sage: e
Elliptic Curve defined by y^2 + y = x^3 - 23737*x + 960366 over Rational Field
sage: e.gens()
[(-171 : 138 : 1), (-647/4 : 6025/8 : 1), (-159 : 845 : 1), (-158 : 875 : 1), (-142 : 1211 : 1), (-136 : 1293 : 1), (-120 : 1442 : 1), (166/9 : -19648/27 : 1)]
sage: e.cremona_label()
'457532830151317a1'

```


What CASes say about the analytic rank of rank 8 elliptic curve '457532830151317a1'
Did you read the Sage's documentation on analytic\_rank()?



```
   Return an integer that is *probably* the analytic rank of this
   elliptic curve.  If leading_coefficient is "True" (only implemented
   for PARI), return a tuple (rank, lead) where lead is the value of
   the first non-zero derivative of the L-function of the elliptic
   curve.

```

So Sage says that it is *probably* 4. This does not qualify as a bug. Further on, the documentation says:



```
   Note: It is an open problem to *prove* that *any* particular elliptic
   curve has analytic rank >= 4.

```

So either the latter is nonsense (well, I am not a number theorist, I don't know), or Magma just solved an open problem for you... Or maybe it didn't, and it fact no CAS can actually compute it.


**EDIT**: please see [this discussion](https://web.archive.org/web/20150709103443/http://permalink.gmane.org/gmane.science.mathematics.categories/2228). Also, [slide 22 of the talk by J.Cremona](http://homepages.warwick.ac.uk/~masgaj/papers/bsd50.pdf) explains that **there is currently no way to check for sure** that the analytic rank $\geq 4$. 


