# What CASes say about the analytic rank of rank 8 elliptic curve '457532830151317a1'

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)
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'

• I thought about voting to close this already some hours ago, but did not then. Now, I do. Mainly, as the last answer further confirms my believe, that it seems better to bring this up on some Sage resource (there are mailing lists and even a Q&A site I believe) right away. Litteraly this question was not on Sage, but then actually it was, and various people on those lists could and I think would also have checked with Magma. The idea to first use MO to test the perceived bug furher and to only then bother Sage devs seems also not good to me. (My vote is OT on 2nd though TL seems better.)
– user9072
Nov 12, 2012 at 18:18
• The real problem with the question is explained in the part of my answer I just added. Basically, there is no algorithm known that will tell you for sure the value in question, if it is at least 4, Nov 13, 2012 at 2:44

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. Also, slide 22 of the talk by J.Cremona explains that there is currently no way to check for sure that the analytic rank $$\geq 4$$.

• Thank you. Interesting discussion slides, and documentation... Didn't mean to be critical of sage, probably will reword the question. The discussion shows some small numbers which might depend on the precision.
– joro
Nov 13, 2012 at 7:26
• Just for your information pari's ellanalyticrank() correctly computed $8$ in 7h, 52min. Looking at sage code, you are using the pari's ellanalyticrank() library function, but almost sure with lower precision -- sage compute $4$ in about 2.4 hours, which is quite faster than pari's default precision.
– joro
Nov 13, 2012 at 14:08
• Oh, I mean pari maybe correctly computed.
– joro
Nov 13, 2012 at 14:10
• right, it's completely open to interpretation. Basically, if you take a positive precision, say, $\epsilon$ and assume that if a number $x$ satisfies $|x|\leq\epsilon$ then $x=0$, then you get your rank to be equal to 8. And if you take precision $10\epsilon$, you get 4, not 8. It actually does not mean that one computation is correct, and the other wrong. Both come with no warranty whatsoever. No monotonicity of any sort. It could still be that with precision $10^{-100}\epsilon$ you would get rank, ray, 5... Nov 13, 2012 at 14:32
• Thank you Dima. You mean a professional human now can't prove if it is 4 or 8 or something else (a human may use better techniques than current CASes)?
– joro
Nov 13, 2012 at 15:22

According to Magma V2.17-5, the analytic rank is $$8$$, as it should be. Below $$s$$ is an approximation of $$\frac{L^{(8)}(1)}{8!}$$.

E:= EllipticCurve([0,0,1,-23737,960366]);

time r, s :=AnalyticRank(E);

Time: 361.950

r;

8

s;

5087.2

Denis Simon's ellrank codes, running in Pari, give rank=8 and the following points of infinite order:

[554, 12563], [-1103/16, 96375/64], [3553/16, 164879/64],

[20233/144, 1090747/1728], [29592/169, 3237242/2197], [34277/169, 4653986/2197],

[47773/361, 2532167/6859], [79874/1849, 9883308/79507]

• Thank you. mwrank finds the generators in seconds too. The question was about the analytic rank :)
– joro
Nov 12, 2012 at 11:59