abvar.fq_isog.adoc

Database report for abvar\fq_isog

Prepared with the lmfdb asciidoc report tool

Christopher Brady <[email protected]>

Version 1.0.0

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Collection information

Description	Isogeny classes of abelian varities over finite fields
Status	beta
Contact	David Roe
Code	abvar/fq

List of keys in the database

The following keys appear in at least one record in the database

Warning

The type of the keys is inferred from only one record in the database. If the key can have different types in different records this type will be inaccurate Only the first 100 characters of the example record are shown

Key	Inferred Type	Example record	Description
A_counts	collection of integer	[1, 577, 195301, 6045229, 891243181, 79220139931L, 4189948510987L, 345796535402469L, 188503571953396 …​	The number of points of the abelian variety over extensions of F_q. Counts are given for $A(F_{q^n})$ for $1 \le n \le max(g,10)$;
angle_numbers	collection of real	[0.3459771436386068, 0.596502288733177]	Frobenius angle numbers. The positive arguments of the roots (considered as complex numbers) of the Weil L-polynomial. There will be g of them unless the list includes 0 or pi.
angle_ranks	integer	2	This is one less than the dimension of the space spanned by the arguments of the roots of the Weil polynomial divided by $\pi$ and one. This might be empty if we haven’t computed it yet.
brauer_invariants	collection of integer stored as string	[u'0', u'0']	The Brauer invariants of the endomorphism algebra. For a simple isogeny class, the number of invariants is the number of primes above p in the number field defined by the Weil polynomial. For a non simple class, the Brauer invariants of its simple factors are concatenated, and they appear in the order in which the factors appear in the field decomposition.
C_counts	collection of integer	[1, 1, 4, 5, -19, 82, 106, 253, 490, 911]	The number of points of a corresponding curve. If the variety is a Jacobian, these are the point counts of a genus g curve of which this is the Jacobian. In particular, if any point counts are negative then this abelian variety cannot be a Jacobian.
decomposition	collection of mixed types	[[u'1.101.a', 1]]	The decomposition into simple factors. The first entry in each pair is the label of the factor, the second is its multiplicity.
g	integer	2	Genus. The degree of the Weil L-polynomial is 2g
galois_n	integer	4	The degree label of the Galois group of the Weil polynomial. If the number field was not in the database when the isogeny class was added to the database, this string is empty. If the isogeny class is not simple, this is also an empty string.
galois_t	integer	3	The transitive label of the Galois group of the Weil polynomial. If the number field was not in the database when the isogeny class was added to the database, this string is empty. If the isogeny class is not simple, this is also an empty string.
known_jacobian	integer	-1	An integer encoding whether the abelian variety is a Jacobian. 1 means that it is definitely a Jacobian, -1 that it is definitely not, and 0 indicates uncertainty.
label	string	1.101.a	LMFDB Label. Labeling Scheme
number_field	string	4.0.16317.1	The label of the number field defined by the Weil polynomial. If the number field was not in the database when the isogeny class was added to the database, this string is empty. If the isogeny class is not simple, this is also an empty string.
p_rank	integer	0	The $p$-rank of the abelian variety. The rank of the $p$-torsion subgroup of the abelian variety. Equal to the number of occurences of the slope 0 in the Newton slopes.
places	collection of mixed types	[[[u'1', u'19/9', u'11/9', u'25/9'], [u'1', u'5/3', u'7/3', u'8/3']]]	The ideals corresponding to the Brauer invariants of the endomorphism algebra. The outer set of lists corresponds to the simple factors of the isogeny class (so in the example, this isogeny class is a product of two simple isogeny classes). For each simple factor, the list contains one list per prime above p in the number field defined by the Weil polynomial. This list describes the prime ideal above p by giving the second generator of the ideal (the first generator is p), as a list of the coefficients of the generator when written in terms of a specific basis for the number field. This basis contains the powers of a root of the P-polynomial (which is the Weil polynomial but reversed)
polynomial	collection of integer	[1, -58, 1263, -12238, 44521]	Coefficients of the Weil L-polynomial. The first entry will always be 1 and the last $q^g$. For i between 0 and g, $a_{2g-i} = q^{g-i} a_i$.
primitive_models	non-primitive type (<type 'list'>)	[]	Every isogeny class defined over smaller fields such that this isogeny class is a base change of this isogeny class. If the isogeny class is primitive, the list is empty. Otherwise, the list contains the label of every primitive isogeny class that base changes to this class. This list is complete.
principally_polarizable	integer	-1	An integer encoding whether the abelian variety is principally polarizable. 1 means that it is definitely principally polarizable, -1 that it is definitely not, and 0 indicates uncertainty.
q	integer	9	Cardinality of Field. All of the roots of the Weil L-polynomial have absolute value $1/\sqrt{q}$.
slopes	collection of integer stored as string	[u'0', u'0', u'1', u'1']	The slopes of the Newton polygon of the Weil polynomial. The slopes are in increasing order, are symmetric under the involution $s \to 1-s$, and the corresponding Newton polygon has endpoints (0,0) and (2g,g).

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List of indices

Index Name	Index fields
polynomial_1	polynomial sorted ascending
decomposition_1	decomposition sorted ascending
p_rank_1	p_rank sorted ascending
C_counts_1	C_counts sorted ascending
id	_id sorted ascending
principally_polarizable_1	principally_polarizable sorted ascending
slopes_1	slopes sorted ascending
label_1	label sorted ascending
known_jacobian_1	known_jacobian sorted ascending
A_counts_1	A_counts sorted ascending

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List of record types in the database

All records

Note	1367543 records of type

• A_counts

• angle_numbers

• angle_ranks

• brauer_invariants

• C_counts

• decomposition

• g

• galois_n

• galois_t

• known_jacobian

• label

• number_field

• p_rank

• places

• polynomial

• primitive_models

• principally_polarizable

• q

• slopes

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Notes

@@abvar\fq_isog\(NOTES)\description@@