# Elliptic Curves
# EllipticCurve([a1,a2,a3,a4,a6]) returns y^2+a1xy+a3y=x^3+a2x^2+a6
# EllipticCurve([a4,a6]) returns y^2=x^3+a4x+a6
# EllipticCurve(j) returns EC with j-invariant j
# EllipticCurve(label) returns the EC over Q from Cremona database
# EllipticCurve(R,[...]) creates EC over the ring R

# Examples of EC/ff and torsion subgroups
E = EllipticCurve([-4, 0]); E

G = E.torsion_subgroup(); G

print "Order:", G.order(), "No. generators:", G.ngens()

for k in range(0,G.ngens()):
    print G.gen(k)

# Example y^2=x^3+2
E = EllipticCurve([0, -2]); E
G = E.torsion_subgroup(); G
print "Order:", G.order(), "No. generators:", G.ngens()
for k in range(0,G.ngens()):
    print G.gen(k)

# Counting the number of points
p=7; n=1; q=p^n;
E=EllipticCurve(GF(q),[0,1]); E
N1=E.cardinality(); N1
E.points()
a=1+q-N1; print "p=", p, "q=", q, "N1(q)=", N1,  "   Defect a=",a

# see also is_quadratic_twist() function

# Weil zeros
# generates Weil Betti polynomial
# lists its roots: Weil zeros

from sage.rings.polynomial.complex_roots import complex_roots # imports the roots function
x=polygen(ZZ); wpoly=1-a*x+q*x^2; wpoly # generates Weil Betti polynomial
wzeros=complex_roots(wpoly, retval='algebraic'); wzeros # computes the roots in alg closure of Q

# Compute the Weil frequencies: wzeros[0]=r e^(it)
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# ... later ...
# define as a routine when stabilized and loop it on a set of primes, 
# n>1 extensions for later ...
︠73a0b496-7b49-464a-b4e6-f8c9a6e95ca4︠