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Package sage :: Package schemes :: Package elliptic_curves :: Module ell_modular_symbols :: Class ModularSymbol
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Class ModularSymbol

source code

                      object --+    
                               |    
structure.sage_object.SageObject --+
                                   |
                                  ModularSymbol


A modular symbol attached to an elliptic curve, which is the map
from $\QQ\to \QQ$ obtained by sending $r$ to the normalized
symmetrized (or anti-symmetrized) integral from r to infinity.

This is as defined in Mazur-Tate-Teitelbaum.  It's possible the
map could be off from what you expect by -1 or +/- 2, but
otherwise it is definitely normalized correctly.

EXAMPLES:


Instance Methods [hide private]
 
__init__(self, E, sign, normalize=True)
INPUT: E -- an elliptic curve sign -- an integer, -1 or 1 normalize -- (default: True); if True, the modular symbol is correctly normalized (up to possibly a factor of -1 or 2).		 source code
 
sign(self)
Return the sign of this elliptic curve modular symbol.		 source code
 
base_ring(self)
Return the base ring for this modular symbol.		 source code
 
elliptic_curve(self)
Return the elliptic curve of this modular symbol...		 source code
 
_call_with_caching(self, r) source code
 
__call__(self, r) source code
 
_repr_(self) source code

Inherited from structure.sage_object.SageObject: __hash__, __new__, __repr__, _axiom_, _axiom_init_, _gap_, _gap_init_, _gp_, _gp_init_, _interface_, _interface_init_, _interface_is_cached_, _kash_, _kash_init_, _macaulay2_, _macaulay2_init_, _magma_, _magma_init_, _maple_, _maple_init_, _mathematica_, _mathematica_init_, _maxima_, _maxima_init_, _octave_, _octave_init_, _pari_, _pari_init_, _r_init_, _sage_, _singular_, _singular_init_, category, db, dump, dumps, plot, rename, reset_name, save, version

Inherited from object: __delattr__, __getattribute__, __reduce__, __reduce_ex__, __setattr__, __str__

Properties [hide private]

Inherited from object: __class__

Method Details [hide private]

__init__(self, E, sign, normalize=True)
(Constructor)

source code  

INPUT:
    E -- an elliptic curve
    sign -- an integer, -1 or 1
    normalize -- (default: True); if True, the modular symbol
        is correctly normalized (up to possibly a factor of
        -1 or 2).  If False, the modular symbol is almost certainly
        not correctly normalized, i.e., all values will be a
        fixed scalar multiple of what they should be.  But
        the initial computation of the modular symbol is
        much faster, though evaluation of it after computing
        it won't be any faster. 

EXAMPLES:
    sage: E=EllipticCurve('11a1')
    sage: import sage.schemes.elliptic_curves.ell_modular_symbols
    sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbol(E,+1)
    sage: M
    Modular symbol with sign 1 over Rational Field attached to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
Overrides: object.__init__

sign(self)

source code  

Return the sign of this elliptic curve modular symbol.

EXAMPLES:
    sage: E=EllipticCurve('11a1')
    sage: import sage.schemes.elliptic_curves.ell_modular_symbols
    sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbol(E,+1)
    sage: M.sign()
    1

base_ring(self)

source code  

Return the base ring for this modular symbol. 
EXAMPLES:
    sage: E=EllipticCurve('11a1')
    sage: import sage.schemes.elliptic_curves.ell_modular_symbols
    sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbol(E,+1)
    sage: M.base_ring()
    Rational Field

elliptic_curve(self)

source code  

Return the elliptic curve of this modular symbol
EXAMPLES:
    sage: E=EllipticCurve('11a1')
    sage: import sage.schemes.elliptic_curves.ell_modular_symbols
    sage: M=sage.schemes.elliptic_curves.ell_modular_symbols.ModularSymbol(E,+1)
    sage: M.elliptic_curve()
    Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field

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