Worksheet_with_APFUNCTION_CANONICAL

307 days ago by Ben.LeVeque

x=var('x') K.<a>=NumberField(x^2-x-1) 
       
from psage.ellcurve.minmodel.sqrt5 import canonical_model def ap(E,p): return E.change_ring(p.residue_field()).trace_of_frobenius() 
       
E = EllipticCurve(K,[a,-1,0,-6*a-1,-7*a-1]) 
       
canonical_model(E.global_minimal_model()).a_invariants() 
        
(a + 1, -a - 1, 0, -33*a - 20, -99*a - 61)
(a + 1, -a - 1, 0, -33*a - 20, -99*a - 61)
E.conductor().norm() 
        
979
979
F = canonical_model(EllipticCurve(K,[0, a - 1, a + 1, -9*a - 8, -21*a - 14]).global_minimal_model()) 
       
F.conductor().norm() 
        
209
209
list(F.a_invariants()) 
        
[0, a, a + 1, -69*a - 43, -372*a - 230]
[0, a, a + 1, -69*a - 43, -372*a - 230]
E.conductor() 
        
Fractional ideal (-6*a - 29)
Fractional ideal (-6*a - 29)
K3 = K.primes_above(3)[0] 
       
ap(E,K3) 
        
-1
-1
E = EllipticCurve(K,[0,a-1,a+1,-9*a-8,-21*a-14]) canonical_model(E.global_minimal_model()).a_invariants() 
        
(0, a, a + 1, -69*a - 43, -372*a - 230)
(0, a, a + 1, -69*a - 43, -372*a - 230)
curve14= EllipticCurve(K,[18*a-28,-105*a+170,-105*a+170,0,0]) 
       
ap(curve14,K.ideal(7)) 
        
8
8
list(primes(100)) 
        
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
71, 73, 79, 83, 89, 97]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
def curves_in_GF(p): ret = {} K = GF(p) for i in range(p): for j in range(p): try: E = EllipticCurve(K,[i,j]) ap = p-len(E.points())+1 if ret.has_key(ap): ret[ap].append(E) else: ret[ap] = [E] except ArithmeticError: pass return ret