1222.sagews

N=38914004
a=RR(N+1-2*sqrt(N))
b=RR(N+1+2*sqrt(N))
a1=a.nearby_rational(max_denominator=1);b1=b.nearby_rational(max_denominator=1)
a1;b1

38901529 38926481

pp=primes(a1+1,b1)
p=pp.next()
p; p.is_prime()

38901529 True

w=RR(sqrt(N))
w1=w.nearby_rational(max_denominator=1)
w1

6238

result=0
for D in range (w1-6200,w1+6200):
        t= -D+4*p
        if( t.is_square()) :
           print t
           print -D

155600676 -5440

D1=-5440
t1=155600676
t2=ZZ(t1.sqrt());
D1.factor();t2

-1 * 2^6 * 5 * 17 12474

hcp=hilbert_class_polynomial(- 2^2 * 5 * 17)
pj=hcp.factor_mod(p)
pj

(x + 17945006) * (x + 18868280) * (x + 26665205) * (x + 27893851)

E1=EllipticCurve(GF(p)(-17945006))
E1

Elliptic Curve defined by y^2 = x^3 + 22185565*x + 38482027 over Finite Field of size 38901529

E1.order(); N; 1+p+t2;1+p-t2

38914004 38914004 38914004 38889056