22.2 Watkins Symmetric Power $ L$ -function Calculator

Module: sage.lfunctions.sympow

SYMPOW is a package to compute special values of symmetric power elliptic curve L-functions. It can compute up to about 64 digits of precision. This interface provides complete access to sympow, which is a standard part of SAGE (and includes the extra data files).

Note: Each call to sympow runs a complete sympow process. This incurs about 0.2 seconds overhead.

Author Log:

ACKNOWLEDGEMENT (from sympow readme):

Class: Sympow

class Sympow
Watkins Symmetric Power $ L$ -function Calculator

Type sympow.[tab] for a list of useful commands that are implemented using the command line interface, but return objects that make sense in SAGE.

You can also use the complete command-line interface of sympow via this class. Type sympow.help() for a list of commands and how to call them.

Functions: analytic_rank,$  $ help,$  $ L,$  $ Lderivs,$  $ modular_degree,$  $ new_data

analytic_rank( self, E)

Return the analytic rank and leading $ L$ -value of the elliptic curve $ E$ .

INPUT:
    E -- elliptic curve over Q

OUTPUT:
    integer -- analytic rank
    string -- leading coefficient (as string)

Note: The analytic rank is not computed provably correctly in general.

Note: In computing the analytic rank we consider $ L^{(r)}(E,1)$ to be 0 if $ L^{(r)}(E,1)/\Omega_E > 0.0001$ .

We compute the analytic ranks of the lowest known conductor curves of the first few ranks:

sage: sympow.analytic_rank(EllipticCurve('11a'))
(0, '2.53842e-01')
sage: sympow.analytic_rank(EllipticCurve('37a'))
(1, '3.06000e-01')
sage: sympow.analytic_rank(EllipticCurve('389a'))
(2, '7.59317e-01')
sage: sympow.analytic_rank(EllipticCurve('5077a'))
(3, '1.73185e+00')
sage: sympow.analytic_rank(EllipticCurve([1, -1, 0, -79, 289]))
(4, '8.94385e+00')
sage: sympow.analytic_rank(EllipticCurve([0, 0, 1, -79, 342]))  # long
(5, '3.02857e+01')
sage: sympow.analytic_rank(EllipticCurve([1, 1, 0, -2582, 48720]))  # long
(6, '3.20781e+02')
sage: sympow.analytic_rank(EllipticCurve([0, 0, 0, -10012, 346900]))  # long
(7, '1.32517e+03')

L( self, E, n, prec)

Return $ L(\Sym ^{(n)}(E,$   edge$ ))$ to prec digits of precision, where edge is the right edge. Here $ n$ must be even.

INPUT:
    E -- elliptic curve
    n -- even integer
    prec -- integer
    
OUTPUT:
    string -- real number to prec digits of precision as a string.

Note: Before using this function for the first time for a given $ n$ , you may have to type sympow('-new_data <n>'), where <n> is replaced by your value of $ n$ .

If you would like to see the extensive output sympow prints when running this function, just type set_verbose(2).

sage: a = sympow.L(EllipticCurve('11a'), 2, 16); a
'1.057599244590958E+00'
sage: RR(a)
1.0575992445909579

Lderivs( self, E, n, prec, d)

Return 0 th to $ d$ th derivatives of $ L(\Sym ^{(n)}(E,s)$ to prec digits of precision, where $ s$ is the right edge if $ n$ is even and the center if $ n$ is odd.

INPUT:
    E -- elliptic curve
    n -- integer (even or odd)
    prec -- integer
    d -- integer
    
OUTPUT:
    a string, exactly as output by sympow

Note: To use this function you may have to run a few commands like sympow('-new_data 1d2'), each which takes a few minutes. If this function fails it will indicate what commands have to be run.

sage: print sympow.Lderivs(EllipticCurve('11a'), 1, 16, 2)
...
 1n0: 2.538418608559107E-01
 1w0: 2.538418608559108E-01
 1n1: 1.032321840884568E-01
 1w1: 1.059251499158892E-01
 1n2: 3.238743180659171E-02
 1w2: 3.414818600982502E-02

modular_degree( self, E)

Return the modular degree of the elliptic curve E, assuming the Stevens conjecture.

INPUT:
    E -- elliptic curve over Q

OUTPUT:
    integer -- modular degree

We compute the modular degrees of the lowest known conductor curves of the first few ranks:

sage: sympow.modular_degree(EllipticCurve('11a'))
1
sage: sympow.modular_degree(EllipticCurve('37a'))
2
sage: sympow.modular_degree(EllipticCurve('389a'))
40
sage: sympow.modular_degree(EllipticCurve('5077a'))
1984
sage: sympow.modular_degree(EllipticCurve([1, -1, 0, -79, 289]))
334976

new_data( self, n)

Pre-compute data files needed for computation of n-th symmetric powers.

Special Functions: __call__,$  $ _curve_str,$  $ _fix_err,$  $ _repr_

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