db-genus2_curves.md

Database genus2_curves

Description	Genus 2 curves over Q
Status	production
Contact	Andrew Sutherland
Code	genus2_curves
Collections	curves, endomorphisms, tamagawa_numbers

Contributors: Andrew Booker, Jeroen Sijsling, Andrew Sutherland, John Voight, and Dan Yasaki

Todo: Match all automorphic friends and move data in temporary Lfunctions.instances collection to Lfunctions database.

Collection curves

• Content: Basic data for genus 2 curves over QQ.
• Contributors: Andrew Sutherland.
• Origin: Code by Andrew Sutherland.
• Extent: 66158 curves so far, see arXiv:1602.03715 for heuristics on the completeness of this collection.

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId	-		assigned by Mongo; contains creation timestamp
aut_grp_id	automorphism group (specified by GAP id)	string encoding list of integers, no spaces	-	u;[6,2]'
class	isogeny class	string	-	u'169.a'
cond	conductor	integer	NN	169
abs_disc	absolute discriminant	integer	NN	169
disc_key	representation of the discriminant D of the curve: the first three digits are floor(log_10 (|D|)), which is followed by digits representing the absolute value of |D|.	string	-	u'002169'
disc_sign	sign of the discriminant	integer	-	-1
g2_inv	G2 invariants	string encoding list of strings representing rationals, no spaces	-	u"['-1/169','33/169','43/169']"
geom_aut_grp_id	geometric automorphism group (shorthand)	string encoding list of 2 integers, no spaces	-	u'[12,4]'
igusa_inv	Igusa invariants	string encoding list of strings representing integers, no spaces	-	u"['1','-33','-43','-283','-169']"
igusa_clebsch_inv	Igusa-Clebsch invariants	string encoding list of strings representing integers	-	u"['8','3172','30056','-692224']"
is_gl2_type	whether the curve is of GL2-type over its base field	boolean	-	True
label	LMFDB label	string	-	u'169.a.169.1'
eqn	coefficients of minimal equation y^2+h(x)y=f(x)	string representing list of 2 lists of integers, no spaces	-	u'[[0,0,0,0,1,1],[1,1,0,1]]
num_rat_wpts	number of rational Weierstrass points	integer	NN_0	0
real_geom_end_alg	endomorphism ring over base field tensored with RR	string	-	u'M_2(R)'
st_group	Sato-Tate group over base field	string	-	u'E_6'
torsion_subgroup	rational torsion group of the Jacobian, represented by the invariant factors [d_1, d_2, ...] for which this torsion group is isomorphic to ZZ / d_1 ZZ x ZZ / d_2 ZZ x ...	string encoding list of integers, no spaces	-	u'[2,2,4]'
torsion_order	rational torsion order of the Jacobian	integer	NN_0	19
two_selmer_rank	2-Selmer rank	integer	NN_0	0
analytic_rank	analytic rank upper bound that is believed to be tight (known for rank 0 or 1)	integer	NN_0	0
has_square_sha	assuming Sha is finite, true if the order of Sha is a square, false otherwise (in which case it is 2 times a square, by a result of Poonen-Stoll 1999)	boolean	-	True
locally_solvable	true if the curve has rational points locally everywhere (i.e. over every completion of Q, including R)	boolean	-	True
globally_solvable	1 if known to have rational points, 0 if known to have no rational points, -1 if unknown	NN	0	True

Index information for curves collection

• abs_disc -- search/browse
• analytic_rank -- search/browse
• aut_grp_id -- search/browse
• class -- list curves in isogeny class
• cond -- search/browse
• cond/class/disc_key/label -- sort order (all ascending)
• disc_key -- search/browse
• g2_inv -- list twists
• geom_aut_grp_id -- search/browse
• has_square_sha -- search/browse
• is_gl2_type -- search/browse
• is_simple_geom -- search/browse
• label -- search
• locally_solvable -- search/browse
• real_geom_end_alg -- search/browse (ST^0)
• st_group -- search/browse
• torsion_order -- search_browse
• torsion_subgroup -- search/browse
• curves.rand (auxilliary collection used for random objection access)

Collection endomorphisms

• Content: Endomorphism data for genus 2 curves over QQ.

• Contributors: Jeroen Sijsling.

• Origin: Code by Jeroen Sijsling

• Extent: Data known for all curves in the genus 2 curves database.

• The representation of the endomorphism lattice by subfields of the full field of definition of the endomorphism ring has a rather terse format. It is a list of lists, and its entries are as follows.

• First entry: A triple that describes the base field by its LMFDB label, a list representing a minimal polynomial, and a list representing a generator in the smallest field over which all endomorphisms are defined, as described by fod_coeffs.

• Second entry: At most two lists that indicate the factors of the endomorphism algebra. Two first entries of these lists base fields of the corresponding factors, as in the description of the first entry above. The third entry indicates whether the corresponding factor is a field or not. If -1 then it is; otherwise this entry is the norm of the discriminant of the corresponding quaternion algebra over the base field described by the first two entries.

• Third entry: A sequence of strings describing End ox RR.

• Fourth entry: A list that describes the endomorphism ring as a subring of the endomorphism algebra. If the second entry is -1, then the first entry gives an index or a conductor norm in the case of a field if that applies. If 0 or 1, then the first entry describes the index of the order and the second entry describes whether it is Eichler (1) or not (0).

• Fifth entry: The Sato-Tate group.

• The conventions above are also used in other fields.

• For splittings of the Jacobian, we return a subfield of smallest degree over which the splitting occurs, represented as above. We also return lists that represent defining equations for the corresponding elliptic curves over that field, or rather, a and b such that the corresponding factor is isomorphic to the curve with equation y^2 = x^3 + (-a/48) x + (-b/864). LMFDB labels for these curves is also return if they exist, and conductor norms are always given.

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
_id	Mongo id	ObjectId	-		assigned my Mongo; contains creation timestamp
factorsQQ_base	description of endomorphism algebra factors over the base field	list of lists	-	[[u'2.0.3.1', [1, -1, 1], -1]]
factorsQQ_geom	description of endomorphism algebra factors over the algebraic closure	list of lists	-	[[u'1.1.1.1', [0, 1], 1]]
factorsRR_base	endomorphism algebra factors over the base field tensored with RR	list of strings	-	[u'CC']
factorsRR_geom	endomorphism algebra factors over the algebraic closure tensored with RR	list of strings	-	[u'M_2(RR)']
fod_coeffs	defining polynomial of the smallest field over which all endomorphisms are defined	list of integers	-	[-1, -3, 6, 4, -5, -1, 1]
fod_label	LMFDB label of the smallest field over which all endomorphisms are defined	string	-	u'6.6.371293.1'
is_simple_base	whether the curve is simple over the base field	boolean	-	True
is_simple_geom	whether the curve is simple over the algebraic closure	boolean	-	False
label	LMFDB label	string	-	u'169.a.169.1'
lattice	endomorphism lattice	list of lists	-	[[[u'1.1.1.1', [0, 1], [u'0', u'0', u'0', u'0', u'0', u'0']], [[u'2.0.3.1', [1, -1, 1], -1]], [u'CC'], [1, -1], u'E_6'], ...]
ring_base	endomorphism ring over the base field as a subring of the endomorphism algebra	list of integers	-	[1, -1]
ring_geom	endomorphism ring over the algebraic closure as a subring of the endomorphism algebra	list of integers	-	[3, 1]
spl_facs_coeffs	defining coefficients of the elliptic curves obtained by splitting the Jacobian	list of lists	-	[[[u'2644', u'9457', u'-8353', u'-15597', u'1720', u'3630'], [u'207207', u'759759', u'-759759', u'-1342341', u'160160', u'316316']]]
spl_facs_condnorms	conductor norms of the elliptic curves obtained by splitting the Jacobian	list of integers	-	[1]
spl_facs_labels	LMFDB labels of the elliptic curves obtained by splitting the Jacobian	list of strings	-	[1]
spl_fod_coeffs	defining polynomial of a field of minimal degree over which a splitting of the Jacobian is defined	list of integers	-	[-1, -3, 6, 4, -5, -1, 1]
spl_fod_gen	generator of a field of minimal degree over which a splitting of the Jacobian is defined, as a subfield of the smallest field over which all endomorphisms are defined	list of strings	-	[u'0', u'1', u'0', u'0', u'0', u'0']
spl_fod_label	LMFDB label of a field of minimal degree over which a splitting of the Jacobian is defined	string	-	u'6.6.371293.1'
st_group_base	Sato-Tate group over the base field	string	-	E_6
st_group_geom	Sato-Tate group over the algebraic closure (equivalently, its identity component)	string	-	E_1
root_number	root number	string	-	u'1'
bad_lfactors	bad primes and the corresponding L-factors	string encoding list of pairs [p,c] where p is a bad prime and c is a list of the coefficients of the Euler factor at p	-	[[13, [1, 5, 13]]]

Index information for endomorphisms collection:

• label -- lookup

Collection tamagaw_numbers

• Content: Tamagawa numbers for bad primes of genus 2 curves over QQ.
• Contributors: Raymond von Bommel.
• Origin: code by Raymond von Bommel.
• Extent: All but 54 tamagaw numbers are present (the missing tamagawa numbers are all for p=2)

Field	Description	Type of stored data	Mathematical type	Example of stored data	Remarks
label	LMFDB label of a genus 2 curve	string	-	169.a.169.1
p	prime of bad reduction for the specified curve	integer	prime number	13
tamagawa_number	tamagawa number of the curve at the prime p	integer	integer	1

Index information for endomorphisms collection:

• label/p - compound index for lookup