db-elliptic_curves.md

Database elliptic_curves

Description	Elliptic curves over Q and other number fields
Status	production
Contact	John Cremona
Code	elliptic_curves
Collections	curves, nfcurves, padic_db

Notes: In good shape over Q (collection curves). Over quadratic and cubic fields (collection nfcurves) complete as far as curves matching Hilbert modular forms go; awaiting addition of curves over more fields of degrees 4-6 to match Hilbert modular forms.

Todo:

• Complete the finding and uploading of curves over totally real fields to match the HMF newforms for fields of degrees >3.
• compute and add p-adic data for curves over Q of conductor > 130000.
• Precompute more data for the nfcurves collection so the webserver does not have to do any nontrivial computations on the fly.

Collection curves

• Content: elliptic curves over Q
• Contributors: John Cremona, Andrew Sutherland, Jeremy Rouse
• Origin: Cremona database, https://github.com/JohnCremona/ecdata
• Extent: complete for conductors up to 380,000 (as of February 2016)

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId	-		assigned my Mongo; contains creation timestamp
label	Cremona label	string	-	'1225a2'
lmfdb_label	LMFDB label	string	-	'1225.a2'
conductor	Conductor	int	N	1225
iso	Cremona isogeny class code	string	-	'11a'
lmfdb_iso	LMFDB isogeny class code	string	-	'11.a'
iso_nlabel	numerical version of the LMFDB isogeny class label	int	Z	0
number	Cremona curve number within its class	int	N	2
lmfdb_number	LMFDB curve number within its class	int	N	2
ainvs	a-invariants (coefficients of minimal reduced Weierstass model)	list of 5 strings representing integers	Z^5	['0', '1', '1', '10617', '75394']
xainvs	a-invariants (coefficients of minimal reduced Weierstass model)	string representing list of 5 integers	Z^5	'[0, 1, 1, 10617, 75394]'
jinv	j-invariant	string representing a rational	Q	'-4096/11'
signD	sign of Discriminant	int	Z	-1	in {-1,+1}
cm	CM code	int	Z	0 (for no CM), or a negative discriminant	in {0, -3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163}
rank	rank	int	N_0	0	May be missing
torsion	torsion order	int	Z	1
torsion_structure	invariants of torsion subgroup	list of at most 2 strings representing ints	N^t (0≤t&le2)	['3']
torsion_generators	generators of torsion subgroup	list of strings representing points	A^2(Q)^t (0≤t&le2)	['(5, 5)']
x-coordinates_of_integral_points	x-coordinates of integral points	string representing list of integers	Z^k	'[5,16]'
gens	generators of infinite order	list of strings representing points	P^2(Q)^k (k≥0)	['(0:0:1)']	May be missing
heights	heights of generators	list of floats	R^k (k≥0)	[0.4754476141654406, 2.4031247402073275]	May be missing
regulator	regulator	float	R	1.0	May be missing; approximate if rank>0
tamagawa_product	Tamagawa product	int	N	4
special_value	special value of r'th derivative of L-function (divided by r!)	float	R	1.490882041449698	approximate
real_period	real period	float	R	0.3727205103624245	approximate
degree	degree of modular parametrization	int	N	1984
non-surjective_primes	primes p for which the mod p Galois representation is not surjective	list of ints	N^k (k≥0)	[5]
galois_images	Sutherland codes for the images of the mod p Galois representations for the non-surjective primes	list of strings	-	['5B']	Sutherland notation; for CM curves, only primes<100
2adic_label	Rouse label of the associated modular curve (None for CM curves)	string	-	'X225g'	based on Rouse, Zureik-Brown classification
2adic_index	index in GL(2,Z2) of the 2-adic representation (or 0 for CM curves)	int	N	1
2adic_log_level	the smallest n such that the image contains the kernel of reduction modulo 2^n (or None for CM curves)	int	N_0	1
2adic_gens	list of matrices in GL(2,Z/2^nZ) generating the image (None for CM curves)	list of lists of 4 ints	GL(2,Z)^k (k≥0)	[[5,0,0,5],[5,5,0,1],[5,5,0,3]]
isogeny_matrix	isogeny matrix	list of lists of ints)	M_k(N) (k≥0)	[[1,5,25],[5,1,5],[25,5,1]]
sha_an	analytic order of Sha	float	R	9.0	approximate unless rank<2
sha	analytic order of sha	int	N	9	rounded value of sha_an
sha_primes	primes dividing sha	list of ints	N^k (k≥0)	[2]
torsion_primes	primes dividing torsion	list of ints	N^k (k≥0)	[2,3]
local_data	reduction data at bad primes	list of dicts, one per prime, each with keys 'p' (value:int), 'ord_cond' (value: int), 'ord_disc' (value: int), 'ord_den_j' (value: int), 'red' (value: int), 'cp' (value: int), 'kod' (value: string)	N	[{'cp': 1, 'kod': '\\( I_{1} \\)', 'ord_cond': 1, 'ord_den_j': 1, 'ord_disc': 1, 'p': 11, 'red': 1}]
min_quad_twist	minimal quadratic twist	dict with keys 'label' (value:string) and 'disc' (value: int)	N	{'disc': 1, 'label': '11a2'}
aplist	Traces of Frobenius	list of 25 ints	Z^25	[0, 1, -1, ..., 2]	a_p for p<100
anlist	L-series coefficients	list of 20 ints	Z^20	[0, 1, -1, ..., 2]	a_n for 0<=np<20

Index information on collection curves:

• {'_id': 1} (created by mongo)
• {'rank': 1, 'number': 1} (for searching and stats)
• {'number': 1} (for searching and stats)
• {'conductor': 1, 'iso_nlabel': 1, 'lmfdb_number': 1} (for sorting)
• {'non-surjective_primes': 1} (for searching)
• {'cm': 1} (for searching)
• {'conductor': 1} (for searching)
• {'lmfdb_label': 1,'number': 1} (for searching)
• {'lmfdb_iso': 1} (for searching)
• {'sha': 1} (for searching)
• {'lmfdb_label': 1} (for searching)
• {'rank': 1} (for searching)
• {'label': 1} (for searching)
• {'jinv': 1} (for searching)
• {'torsion_structure': 1} (for searching)
• {'iso': 1} (for searching)
• {'torsion': 1} (for searching)
• {'label': 1, 'number': 1} (for searching)
• {'xainvs': 1} (for searching)
• curves2.rand (auxilliary collection used for random objection access)

Collection padic_db

• Content: p-adic regulators for elliptic curves over Q
• Contributors: unkown
• Origin: unknown
• Extent: primes p with 3,p<100 of good ordinary reduction, for curves of conductor up to 130,000 only (last updated in 2010)

-

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId			assigned my Mongo; contains creation timestamp
lmfdb_iso	LMFDB label of isogeny class	string	-	'58.a'
p	prime	int	N (prime)	97
prec	p-adic precision	int	N	20
unit	unit factor of regulator	string representing integer	Z_p (mod p^N)	'8471152617139064438417376357679138234'
val	valuation of p-adic regulator	int	N_0	1

Collection nfcurves

• Content: elliptic curves over number fields other than Q

• Contributors: John Cremona, Alyson Deines, Steve Donelly, Paul Gunnells, Warren Moore, Haluk Sengun, John Voight, Dan Yasaki.

• Origin: https://github.com/JohnCremona/ecnf-data

• Extent: contains curves over several totally real fields (of degrees up to 6) and a few imaginary quadratic fields, in each case complete up to some conductor norm bound

• Note: This collection is currently being rebuilt with additional data fields; the new version is temporarily called nfcurves2. When this process is complete and the necessary code changes made the description here will be updated, and index information added.

• Explanation of data fields representing elements of the field, including points:

  • Each field of degree d has a distinguished generator
  • A rational number is represented as a string
  • A field element is represented as a list of d strings, each representing the coordinate with respect to the power basis
  • A point is represented (in projective coordinates) as a list of 3 field elements, i.e. a list of 3 lists of d strings
  • Generator fields are represented by lists of points

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId	-		assigned my Mongo; contains creation timestamp
field_label	Base field label	string		'2.0.8.1'
degree	Base field degree	int	N	2
signature	Base field signature	list of 2 ints	N_0^2	[0, 1]
abs_disc	absolute value of discriminant of base field	int	N	8
label	full label	string	-	'2.0.8.1-[3618,1146,3]-e2'
short_label	short label (excludes field)	string	-	'[3618,1146,3]-e2'
class_label	full label of isogeny class	string	-	'2.0.8.1-[3618,1146,3]-e'
short_class_label	short label of isogeny class (excludes field)	string	-	'2.0.8.1-[3618,1146,3]-e'
conductor_label	condcutor label	string	-	'[3618,1146,3]' or '37.1'
iso_label	isogeny class label	string	-	'e'	base 26 representation of isogeny class index
iso_nlabel	isogeny class index	int	N_0	4
conductor_ideal	data defining the conductor ideal	string	-	'[3618,1146,3]'	representation generators
conductor_norm	conductor norm	int	N	3618
number	index of curve in isogeny class	int	N	2	starts at 1
isogeny_matrix	Isogeny matrix	list of list of ints (degrees)	M_k(N) (k≥0)	[[1, 2], [2, 1]]
ainvs	a-invariants	list of 5 lists of d strings		[['1', '0'], ['1', '-1'], ['1', '1'], ['17', '-7'], ['9', '22']]	coordinates with respect to power basis; d is the degree of the field
jinv	j-invariant	list of d strings		['288857903821/4771277298', '7556047939133/9542554596']	coordinates with respect to power basis; d is the degree of the field
analytic_rank	analytic rank	int	N_0	0
rank	rank	int	N_0	0
rank_bounds	lower and upper rank bounds	list of 2 ints	N_0^2	[0, 0]
sha_an	analytic order of Sha	int	N	1	rounded float
gens	generators of infinite order	list of lists of 3 lists of d strings	A^2(K)^r (0≤r≤rank)	[[['2', '2'], ['11', '-12'], ['1', '0']]]
torsion_order	torsion order	int	N	6
torsion_structure	invariants of torsion subgroup	list of at most 2 ints	N^t (0≤t&le2)	[3, 3]
torsion_gens	torsion generators	list of lists of 3 lists of d strings	A^2(K)^t (0≤t&le2)	[[['-9', '2'], ['16', '18'], ['1', '0']]]
q_curve	Q-curve flag	boolean	{True, False}	False
base_change	labels of base change source curves	list of strings	-	['4032.k2', '63.a2']
cm	CM code	int	Z	0 (for no CM), or a negative discriminant

Index information for collection nfcurves (currently incomplete, to be added)

• nfcurve.rand (auxilliary collection used for random objection access)