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
¶ Bases:
sage.structure.sage_object.SageObject
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 curven
- even integerprec
- 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')
, wheren
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 curven
- integer (even or odd)prec
- integerd
- 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 rankstring
- 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.
-