# Re: [sage-support] points on a Fermat curve

```On Fri, Apr 13, 2018 at 7:26 AM, John Cremona <john.crem...@gmail.com> wrote:
> This looks like a bug to me:
>
> sage: F=GF(3)
> sage: R.<X,Y,Z>=F[]
> sage: C=Curve(X^8+Y^8-Z^8)
> sage: C.count_points(1) # correct count over GF(3^1)
> [4]
> sage: C.count_points(8) # should give counts over GF(3^n) for n=1..8 but it
> crashes
>
> TypeError: F (=[X^8 + Y^8 - Z^8]) must be a list or tuple of polynomials of
> the coordinate ring of A  (=Projective Space of dimension 2 over Finite
> Field in z2 of size 3^2)
>
> -- note that the error message looks suspicious.
>
> Also the more naive
>
> sage: F=GF(3^8)
> sage: R.<X,Y,Z>=F[]
> sage: C=Curve(X^8+Y^8-Z^8)
> sage: C.count_points(1)
> [4696]
>
> works but is so slow it must be doing something very basic such as testing
> every point in the plane (of which there are 3^16+3^8+1...)
>
> John
>```
```
Agreed.

Moreover, according to
http://doc.sagemath.org/html/en/constructions/algebraic_geometry.html,
"The option algorithm="bn uses Sage’s Singular interface and calls the
brnoeth package."
However, when I look at the doc string in C.rational_points? it says the option
algorithm is ignored. Times seem to bear this out:

sage: x,y,z = PolynomialRing(GF(3^7), 3, 'xyz').gens()
sage: C = Curve(z^8 - y^8 - x^8)
sage: time Cpts = C.rational_points()
CPU times: user 10.9 s, sys: 243 ms, total: 11.1 s
Wall time: 10.9 s
sage: time Cpts = C.rational_points(algorithm="bn")
CPU times: user 10.8 s, sys: 182 ms, total: 11 s
Wall time: 10.8 s
sage: x,y,z = PolynomialRing(GF(3^8), 3, 'xyz').gens()
sage: C = Curve(z^8 - y^8 - x^8)
sage: time Cpts = C.rational_points()
CPU times: user 22.8 s, sys: 97.2 ms, total: 22.9 s
Wall time: 22.8 s
sage: time Cpts = C.rational_points(algorithm="bn")
CPU times: user 22.8 s, sys: 97.2 ms, total: 22.9 s
Wall time: 22.9 s

>
> On 13 April 2018 at 11:55, David Joyner <wdjoy...@gmail.com> wrote:
>>
>> Hi:
>>
>> The question below is posted for Gary McGuire, who is not a subscriber
>> to this list:
>>
>> "I would like to know the number of rational points on the (projective)
>> curve
>> x^8+y^8=z^8
>> over the field of order 3^{18}.
>>
>> My question is, can Sage do this calculation, and how?"
>>
>> - David
>>
>> PS: About 3 years ago, a related question was posted:
>> For some reason, the method described there is no longer implemented.
>>
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>
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