EtaFunction,c

328 days ago by Ben.LeVeque

Curve [1,a+1,a,a,0]

y^2 + xy + ay = x^3 + (a+1)x^2 + ax

b=a^10*47*(a+5);b 
        
17108*a + 10575
17108*a + 10575
K.<a>=NumberField(x^2-x-1) embs=K.embeddings(CC) def eta(alpha): return (alpha*alpha).trace() def c(alpha): while True: plus=a*alpha minus=alpha/a if eta(plus)<eta(alpha): alpha=plus continue if eta(minus)<eta(alpha): alpha=minus continue return alpha 
       
x = var('x') K.<a> = NumberField(x^2-x-1) E = EllipticCurve(K,[1,a+1,a,a,0]) 
       
norm(-4*a+1) 
        
-19
-19
K.factor(209) 
        
(Fractional ideal (3*a - 2)) * (Fractional ideal (3*a - 1)) *
(Fractional ideal (-4*a + 1)) * (Fractional ideal (-4*a + 3))
(Fractional ideal (3*a - 2)) * (Fractional ideal (3*a - 1)) * (Fractional ideal (-4*a + 1)) * (Fractional ideal (-4*a + 3))
b=a^10*(3*a+15) 
       
timeit('c(b)') 
        
625 loops, best of 3: 135 µs per loop
625 loops, best of 3: 135 µs per loop
print 'Is c(b*b)=c(b)*c(b)? ', c(b*b)==c(b)*c(b) print 'c(b): ', c(b) print 'c(b)^2: ', c(b)*c(b) print 'c(b^2): ', c(b*b) 
        
Is c(b*b)=c(b)*c(b)?  False
c(b):  3*a + 15
c(b)^2:  99*a + 234
c(b^2):  234*a - 135
Is c(b*b)=c(b)*c(b)?  False
c(b):  3*a + 15
c(b)^2:  99*a + 234
c(b^2):  234*a - 135
K.<sqrt5>=QuadraticField(5); K 
        
Number Field in sqrt5 with defining polynomial x^2 - 5
Number Field in sqrt5 with defining polynomial x^2 - 5
K.embeddings(CC) 
        
[
Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.618033988749895,
Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.61803398874989
]
[
Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.618033988749895,
Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 1.61803398874989
]
sqrt5.charpoly() 
        
x^2 - 5
x^2 - 5
K.ring_of_integers().basis() 
        
[1/2*sqrt5 + 1/2, sqrt5]
[1/2*sqrt5 + 1/2, sqrt5]
x=var('x') K.<a>=NumberField(x^2-x-1);K 
        
Number Field in a with defining polynomial x^2 - x - 1
Number Field in a with defining polynomial x^2 - x - 1
a.charpoly() 
        
x^2 - x - 1
x^2 - x - 1
K.ring_of_integers().basis() 
        
[1, a]
[1, a]
K.disc() 
        
5
5
E=EllipticCurve([1,a+1,a,a,0]);E 
        
Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over
Number Field in a with defining polynomial x^2 - x - 1
Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
show(E) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + xy + ay = x^3 + \left(a + 1\right)x^2 + ax 
\newcommand{\Bold}[1]{\mathbf{#1}}y^2 + xy + ay = x^3 + \left(a + 1\right)x^2 + ax
N=E.conductor();N 
        
Fractional ideal (5*a - 2)
Fractional ideal (5*a - 2)

This ^ means that the conductor is

5a + 2
alpha=N.gens_reduced()[0];alpha 
        
5*a - 2
5*a - 2
alpha.norm() 
        
-31
-31
E.torsion_order() 
        
8
8
T=E.torsion_subgroup();T 
        
Torsion Subgroup isomorphic to Z/8 associated to the Elliptic Curve
defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in
a with defining polynomial x^2 - x - 1
Torsion Subgroup isomorphic to Z/8 associated to the Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
T.invariants() 
        
(8,)
(8,)
E 
        
Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over
Number Field in a with defining polynomial x^2 - x - 1
Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
embs = K.embeddings(RR);embs 
        
[
Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Real Field with 53 bits of precision
  Defn: a |--> -0.618033988749895,
Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Real Field with 53 bits of precision
  Defn: a |--> 1.61803398874989
]
[
Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Real Field with 53 bits of precision
  Defn: a |--> -0.618033988749895,
Ring morphism:
  From: Number Field in a with defining polynomial x^2 - x - 1
  To:   Real Field with 53 bits of precision
  Defn: a |--> 1.61803398874989
]
 
       
E.period_lattice(embs[0]).basis() 
        
(3.05217315335726, 2.39884476932372*I)
(3.05217315335726, 2.39884476932372*I)

^ This implies that the real period is the real part of ^ multiplied by 2. If the lattice does not have a rectangular fundamental domain, then it is just the real part.
E.j_invariant() 
        
-106208/31*a + 51455/31
-106208/31*a + 51455/31
E.b_invariants() 
        
(4*a + 5, 3*a, a + 1, a)
(4*a + 5, 3*a, a + 1, a)
E.arithmetic_genus() 
        
1
1
E.automorphisms() 
        
[Generic endomorphism of Abelian group of points on Elliptic Curve
defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in
a with defining polynomial x^2 - x - 1
  Via:  (u,r,s,t) = (1, 0, 0, 0), Generic endomorphism of Abelian group
of points on Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2
+ a*x over Number Field in a with defining polynomial x^2 - x - 1
  Via:  (u,r,s,t) = (-1, 0, -1, -a)]
[Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
  Via:  (u,r,s,t) = (1, 0, 0, 0), Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 + x*y + a*y = x^3 + (a+1)*x^2 + a*x over Number Field in a with defining polynomial x^2 - x - 1
  Via:  (u,r,s,t) = (-1, 0, -1, -a)]
E.genus() 
        
1
1
E.tamagawa_numbers() 
        
[1]
[1]
E.j_invariant? 
       

File: /sagenb/sage_install/sage-4.7/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_generic.py

Type: <type ‘instancemethod’>

Definition: E.j_invariant()

Docstring:

Returns the j-invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.j_invariant()
110592/37
sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.j_invariant()
-122023936/161051
sage: E = EllipticCurve([-4,0])
sage: E.j_invariant()
1728

sage: E = EllipticCurve([GF(7)(2),1])
sage: E.j_invariant()
1

File: /sagenb/sage_install/sage-4.7/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_generic.py

Type: <type ‘instancemethod’>

Definition: E.j_invariant()

Docstring:

Returns the j-invariant of this elliptic curve.

EXAMPLES:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.j_invariant()
110592/37
sage: E = EllipticCurve([0, -1, 1, -10, -20])
sage: E.j_invariant()
-122023936/161051
sage: E = EllipticCurve([-4,0])
sage: E.j_invariant()
1728

sage: E = EllipticCurve([GF(7)(2),1])
sage: E.j_invariant()
1
K.prime_above(31) 
        
Fractional ideal (5*a - 2)
Fractional ideal (5*a - 2)
E.kodaira_symbol(5*a-2) 
        
I1
I1
E=EllipticCurve(K,'11a') 
       
E.Np(5*a-2) 
        
Traceback (click to the left of this block for traceback)
...
AttributeError: 'EllipticCurve_number_field' object has no attribute
'Np'
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_55.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("RS5OcCg1KmEtMik="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpaudx_2/___code___.py", line 3, in <module>
    exec compile(u'E.Np(_sage_const_5 *a-_sage_const_2 )
  File "", line 1, in <module>
    
  File "parent.pyx", line 738, in sage.structure.parent.Parent.__getattr__ (sage/structure/parent.c:5754)
  File "parent.pyx", line 177, in sage.structure.parent.raise_attribute_error (sage/structure/parent.c:2726)
AttributeError: 'EllipticCurve_number_field' object has no attribute 'Np'
K.unit_group() 
        
Unit group with structure C2 x Z of Number Field in a with defining
polynomial x^2 - x - 1
Unit group with structure C2 x Z of Number Field in a with defining polynomial x^2 - x - 1
E=EllipticCurve(K,[2,3,4,a,1]) 
       
E 
        
Elliptic Curve defined by y^2 + 2*x*y + 4*y = x^3 + 3*x^2 + a*x + 1 over
Number Field in a with defining polynomial x^2 - x - 1
Elliptic Curve defined by y^2 + 2*x*y + 4*y = x^3 + 3*x^2 + a*x + 1 over Number Field in a with defining polynomial x^2 - x - 1
N=E.conductor();N 
        
Fractional ideal (5*a - 2)
Fractional ideal (5*a - 2)
N.gens_reduced()[0].norm() 
        
-31
-31
N.norm() 
        
31
31
dumps? 
       

File: /sagenb/sage_install/sage-4.7/devel/sage/sage/structure/sage_object.pyx

Type: <type ‘builtin_function_or_method’>

Definition: dumps(obj, compress=True)

Docstring:

Dump obj to a string s. To recover obj, use loads(s).

See also

dumps()

EXAMPLES:

sage: a = 2/3
sage: s = dumps(a)
sage: print len(s)
49
sage: loads(s)
2/3

File: /sagenb/sage_install/sage-4.7/devel/sage/sage/structure/sage_object.pyx

Type: <type ‘builtin_function_or_method’>

Definition: dumps(obj, compress=True)

Docstring:

Dump obj to a string s. To recover obj, use loads(s).

See also

dumps()

EXAMPLES:

sage: a = 2/3
sage: s = dumps(a)
sage: print len(s)
49
sage: loads(s)
2/3