# Watkins Symmetric Power $$L$$-function Calculator¶

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.

AUTHORS:

• Mark Watkins (2005-2006): wrote and released sympow
• William Stein (2006-03-05): wrote Sage interface

ACKNOWLEDGEMENT (from sympow readme):

• The quad-double package was modified from David Bailey’s package: http://crd.lbl.gov/~dhbailey/mpdist/
• The squfof implementation was modified from Allan Steel’s version of Arjen Lenstra’s original LIP-based code.
• The ec_ap code was originally written for the kernel of MAGMA, but was modified to use small integers when possible.
• SYMPOW was originally developed using PARI, but due to licensing difficulties, this was eliminated. SYMPOW also does not use the standard math libraries unless Configure is run with the -lm option. SYMPOW still uses GP to compute the meshes of inverse Mellin transforms (this is done when a new symmetric power is added to datafiles).
class sage.lfunctions.sympow.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.

L(E, n, prec)

Return $$L(\mathrm{Sym}^{(n)}(E, \text{edge}))$$ 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).

EXAMPLES:

These examples only work if you run sympow -new_data 2 in a Sage shell first. Alternatively, within Sage, execute:

sage: sympow('-new_data 2')  # not tested


This command precomputes some data needed for the following examples.

sage: a = sympow.L(EllipticCurve('11a'), 2, 16)  # not tested
sage: a                                          # not tested
'1.057599244590958E+00'
sage: RR(a)                                      # not tested
1.05759924459096

Lderivs(E, n, prec, d)

Return $$0^{th}$$ to $$d^{th}$$ derivatives of $$L(\mathrm{Sym}^{(n)}(E,s)$$ 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.

EXAMPLES:

sage: print(sympow.Lderivs(EllipticCurve('11a'), 1, 16, 2))  # not tested
...
1n0: 2.538418608559107E-01
1w0: 2.538418608559108E-01
1n1: 1.032321840884568E-01
1w1: 1.059251499158892E-01
1n2: 3.238743180659171E-02
1w2: 3.414818600982502E-02

analytic_rank(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 $$L^{(r)}(E,1)/\Omega_E > 0.0001$$.

EXAMPLES: 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 time
(5, '3.02857e+01')
sage: sympow.analytic_rank(EllipticCurve([1, 1, 0, -2582, 48720]))  # long time
(6, '3.20781e+02')
sage: sympow.analytic_rank(EllipticCurve([0, 0, 0, -10012, 346900]))  # long time
(7, '1.32517e+03')

help()
modular_degree(E)

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

INPUT:

• E - elliptic curve over Q

OUTPUT:

• integer - modular degree

EXAMPLES: 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(n)

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