Francois Morain on Wed, 20 Jan 1999 17:39:34 +0100

 Bug in ellap (?)

```Hi folks,

I have found a strange thing about ellap. As is well known, when
E:Y^2=X^3+aX with p = 3 mod 4, then ap = 0. Yet, trying the following:

{bug(nmin, nmax, p)=
for(n = nmin, nmax,
p = 10^n+3;
while(isprime(p) == 0, p = p+4);
print("n=", n, " p=", p, " -> ",ellap(ellinit([0,0,0,2,0]),p));
);
}

I got:

GP/PARI CALCULATOR Version 2.0.13 (alpha)
Alpha 64-bit version
(readline disabled, extended help not available)

C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

realprecision = 38 significant digits
seriesprecision = 16 significant terms
format = g0.38

parisize = 8000000, primelimit = 500000
? bug(2, 20)
n=2 p=103 -> 0
n=3 p=1019 -> 0
n=4 p=10007 -> 0
n=5 p=100003 -> 0
n=6 p=1000003 -> 0
n=7 p=10000019 -> 0
n=8 p=100000007 -> 0
n=9 p=1000000007 -> 0
n=10 p=10000000019 -> 1
n=11 p=100000000003 -> -1
n=12 p=1000000000039 -> -1
n=13 p=10000000000051 -> -1
n=14 p=100000000000031 -> 1
n=15 p=1000000000000091 -> 1
n=16 p=10000000000000079 -> 1
n=17 p=100000000000000003 -> -1
n=18 p=1000000000000000003 -> -1
n=19 p=10000000000000000051 -> -1
n=20 p=100000000000000000039 -> -1

What happened?

Since I am curious, I tried with Y^2=X^3+b and p=2 mod 3:

n=2 p=311 -> 0
n=3 p=3011 -> 0
n=4 p=30011 -> 0
n=5 p=300017 -> 0
n=6 p=3000017 -> 0
n=7 p=30000023 -> 0
n=8 p=300000047 -> 0
n=9 p=3000000077 -> -1
n=10 p=30000000053 -> -1
n=11 p=300000000077 -> -1
n=12 p=3000000000167 -> 1
n=13 p=30000000000011 -> -1
n=14 p=300000000000089 -> 1
n=15 p=3000000000000041 -> 1
n=16 p=30000000000000029 -> -1
n=17 p=300000000000000011 -> -1
n=18 p=3000000000000000059 -> -1
n=19 p=30000000000000000041 -> 1
n=20 p=300000000000000000053 -> -1

Any idea? Related to p getting too large (> 2^31)???

F. Morain
```