postgres_schema.md

Table name: av_fq_isog

This table represents unpolarized abelian varieties, up to isogeny.

Column	Type	Notes
label	text
g	smallint
q	integer
poly_str	text	Space separated string of coefficients, for searching
p_rank	smallint
dim1_factors	smallint	Number of dimension 1 factors
dim2_factors	smallint	Number of dimension 2 factors
dim3_factors	smallint	Number of dimension 3 factors
dim4_factors	smallint	Number of dimension 4 factors
dim5_factors	smallint	Number of dimension 5 factors
dim1_distinct	smallint	Number of distinct dimension 1 factors
dim2_distinct	smallint	Number of distinct dimension 2 factors
dim3_distinct	smallint	Number of distinct dimension 3 factors
dim4_distinct	smallint	Number of distinct dimension 4 factors
dim5_distinct	smallint	Number of distinct dimension 5 factors
geom_dim1_factors	smallint	Number of dimension 1 factors
geom_dim2_factors	smallint	Number of dimension 2 factors
geom_dim3_factors	smallint	Number of dimension 3 factors
geom_dim4_factors	smallint	Number of dimension 4 factors
geom_dim5_factors	smallint	Number of dimension 5 factors
geom_dim1_distinct	smallint	Number of distinct dimension 1 factors
geom_dim2_distinct	smallint	Number of distinct dimension 2 factors
geom_dim3_distinct	smallint	Number of distinct dimension 3 factors
geom_dim4_distinct	smallint	Number of distinct dimension 4 factors
geom_dim5_distinct	smallint	Number of distinct dimension 5 factors
poly	integer[]	Coefficients of the Weil polynomial. The first will always be 1 and the last q^g
real_poly	integer[]	Coefficients of the real Weil polynomial, whose roots are the traces down to R of the roots of the Weil polynomial
angles	float8[]	Angles corresponding to roots in the closure of the upper half plane, divided by pi. All will be in the interval [0, 1], and there will be g of them unless 0 or 1 is included.
angle_rank	smallint	The dimension of the Q-span of the angles (see knowl for complete definition)
slopes	text[]	The sorted list of slopes, as string representations of rational numbers. Duplicated slopes will have "A", "B", etc appended.
abvar_counts	numeric[]	The list of counts #A(F_{q^i}) for i=1..10, for A in this isogeny class
abvar_counts_str	text	A space separated string of abelian variety counts, for searching
abvar_count	numeric	The count #A(F_q), duplicated for searching purposes
curve_counts	numeric[]	The list of curve counts #C(F_{q^i}) for i=1..10 for any curve C of genus g with J(C) in this isogeny class
curve_counts_str	text	A space separated string of curve counts, for searching
curve_count	integer	The count #C(F_q), duplicated for searching purposes
has_jacobian	smallint	1 if it is known that this isogeny class contains a Jacobian; -1 if it is known that it does not; 0 otherwise
has_principal_polarization	smallint	1 if it is known that this isogeny class contains a principally polarizable abelian variety; -1 if it is known that it does not; 0 otherwise
is_simple	boolean
is_geometrically_simple	boolean
is_supersingular	boolean
simple_factors	text[]	A list of labels of simple factors. Duplicated factors will have "A", "B", etc appended.
simple_distinct	text[]	A list of distinct labels of simple factors.
simple_multiplicities	smallint[]	For each distinct simple factor, the multiplicity in the decomposition.
number_fields	text[]	The number fields associated to the irreducible factors of the Weil polynomial
galois_groups	text[]	The Galois groups of the number fields associated to the irreducible factors of the Weil polynomial, e.g. "4T3"
geometric_extension_degree	smallint	The smallest degree extension of the base field over which the endomorphism algebra becomes the full endomorphism algebra
center_dim	smallint	The dimension of the center of the endomorphism algebra End^0(A) over Q
geometric_center_dim	smallint	The dimension of the center of the geometric endomorphism algebra End^0_{q^r}(A) over Q, where r is the geometric extension degree
primitive_models	text[]	A list of labels giving primitive models for this isogeny class (ie, this class arises from base change from the model). If primitive, NULL.
is_primitive	boolean
twists	jsonb	A list of triples (label, geom_label, r) where label is the label of a twist, r is an extension degree where the twists become isomorphic, and geom_label is the label of the common base change to that degree.
size	integer	number of isomorphism classes within the isogeny class (isomorphisms of unpolarized abelian varieties)
ppav_count	integer	number of isomorphism classes of principally polarized abelain varieties within the isogeny class (isomorphisms of polarized abelian varieties)
jacobian_count	integer	number of isomorphism classes of Jacobians within the isogeny class (isomorphisms of polarized abelian varieties)
zfv_is_bass	boolean	whether all the over-orders for the order Z[F,V] are Gorenstein
zfv_is_maximal	boolean	whether the order Z[F,V] is maximal
zfv_index	numeric	the index of the order Z[F,V] in the maximal order
zfv_index_factorization	numeric[]	A list of pairs (p, e) giving the factorization of the index
zfv_plus_index	numeric	the index of the order Z[F+V] in the maximal order of the real subfield
zfv_plus_index_factorization	numeric[]	A list of pairs (p, e) giving the factorization of the index
zfv_plus_norm	numeric	The absolute value of the norm of F-V to Z
isogeny_graphs	jsonb	list of pairs (p, G), where p is a degree (or maybe list of degrees) and G is a list of pairs (u,v) representing the directed edge from u to v. Each of u and v is the isom_letter for the corresponding isogeny class
ideal_class_generators	text[]	A list of isom_letters for isomorphism classes that generate the ideal monoid
ideal_class_relations	integer[]	A matrix of positive integers giving relations between the ideal class generators
cm_type	boolean[]	Whether the +imaginary embedding is a p-adic non-unit, for embeddings sorted by real part
cm_elt	numeric[]	An element of Q[F] that is positive imaginary under each embedding in the CM type
curves	text[]	A list of curves in this isogeny class
hyp_count	integer	The number of isomorphism classes of Jacobians of hyperelliptic curves within this isogeny class (isomorphisms of polarized abelian varieties)

Table name: av_fq_endalg_factors

There will be a row in this table for each simple factor of an isogeny class over an extension field GF(q^r), where r divides the geometric extension degree

Column	Type	Notes
base_label	text	The label for the base isogeny class, with (q,g) in our range, either simple or non-simple
extension_label	text	The label for a simple factor of the base change
extension_degree	smallint	The degree of the extension (could be 1)
multiplicity	smallint	The multiplicity of this simple factor in the base change

Table name: av_fq_endalg_data

Data to specify endomorphism algebra for base changed isogeny classes, as a division algebra over its center

Column	Type	Notes
extension_label	text	The label for the base changed simple isogeny class (which may be out of our (g,q) range)
center	text	The number field label for the center of the endomorphism algebra End^0_{q^r}(A)
galois_group	text	The transitive label (e.g. "4T3") for the Galois group of the center
center_dim	smallint	The degree of the center over Q
divalg_dim	smallint	The dimension of the endomorphism algebra End^0_{q^r}(A) over its center
places	text	A list of lists of rational numbers stored as strings, giving the prime ideals above p. The terms in the outer list correspond to places in the corresponding number field, and each inner list gives coefficients for a two-element generator of that prime ideal (along with p) as coefficients of powers of F. They are sorted so that the valuation of F is increaasing.
brauer_invariants	text[]	A list of rational numbers stored as strings, giving the Brauer invariants for End^0_{q^r}(A) as a division algebra over its center

Table name: av_fq_weak_equivalences

Representatives for the weak equivalence classes

Column	Type	Notes
label	text
we_number	smallint	enumeration of the weak equivalence classes within a given isogeny class
pic_size	integer	Size of Pic(S)
multiplicator_ring	text	label for the multiplicator ring S
isog_label	text	label for the isogeny class
ideal_basis_numerators	numeric[]	Z-basis for the chosen representative of weak equivalence class, after scaling by the denominator
ideal_basis_denominator	numeric	denominator for coefficients in the Z-basis (will be a divisor of the index of the Frobenius order in the maximal order)
is_invertible	boolean	Invertible in its multiplicator ring
is_gorenstein	boolean	Whether the multiplicator ring is Gorenstein
inverting_element	numeric[]	When invertible, an element x so that I/x is the ring (null if not invertible), expressed in terms of V^g,...,V,1,F,F^2,...,F^{g-1}
minimal_overorders	smallint[]	list of we_numbers for minimal overorders

Table name: av_fq_pic

Rows give representatives for generators of Pic(S) as S ranges over multiplicator rings

Column	Type	Notes
generator_number	smallint	Which generator
multiplicator_ring	text	Label for the multiplicator ring S in the weak equivalence classes table
ideal_basis_numerators	numeric[]	Z-basis for this ideal, after scaling by the denominator
ideal_basis_denominator	numeric	denominator for coefficients in the Z-basis (will be a divisor of the index of the Frobenius order in the multiplicator ring)
multiplicative order	integer	multiplicative order in Pic(S)

Table name: av_fq_isom

This table represents unpolarized abelian varieties, up to isomorphism. It would be nice to give the lattice of polarizations (dimension should be the rank of the endomorphism algebra over the base field). Minimal polarization degree is generalization of can_be_principally_polarized. Are there local invariants (of the weak equivalence class) that have consequences for the abelian variety?

Column	Type	Notes
label	text	g.q.weil.enum, where g is the dimension, q is the cardinality of the base field, weil is the encoding of the Weil polynomial and enum is isom_letter
isom_num	integer	A 0-based enumeration of the isomorphism classes within an isogeny class, TBD
isom_letter	text	Base 26 a-z encoding of isom_num
isog_label	text	label for isogeny class
isog_power	smallint	When the Weil polynomial is h^r for some squarefree polynomial h, we record r. If r > 1, require h to be Bass (unable to compute otherwise)
weak_equivalence_class	smallint	The we_number for the row in the weak equivalence class table
endo_ring	smallint	The we_number for the row in the weak equivalence class table corresponding to the endomorphism ring (NULL when isog_power != 1)
rep_type	smallint	0=ordinary or Centeleghe-Stix,...
rational_invariants	numeric[]	Invariant factors of A(F_q)
is_product	boolean	Whether this isomorphism class is a product of smaller dimensional abelian varieties
power_product_factorization	text[]	If isog_power > 1, list of isom_letters corresponding to a direct sum decomposition S_1 + S_2 + ... + I_r in the isogeny class corresponing to the polynomial h. Here S_i is an order and (I_r:I_r) >= S_{r-1}.
product_factorization	jsonb	List of pairs (label, e) expressing this as a product of smaller dimensional abelian varities (NULL if not)
related_objects	text[]	List of URLs (null for now)
principal_polarizations	smallint	The number of principal polarizations (null if unknown)
is_reduced	boolean	Whether the fractional ideal is reduced (HNF, minimal norm, lexicographic within same norm) (add later)

Table name: av_fq_pol

This table represents polarized abelian varieties, up to isomorphism.

Column	Type	Notes
label	text	?????
isom_label	text
representative
degree	smallint	degree of the polarization
kernel	smallint[]	invariant factors for the kernel of the isogeny (cokernel of the map of lattices)
is_decomposable	boolean	Whether this polarized abelian variety is a product
decomposition	jsonb	List of pairs (label, e) expressing this polarized abelian variety as a product (NULL if not)
aut_group	text	GAP id
geom_aut_group	text	GAP id
is_hyperelliptic	boolean
is_geometrically_hyperelliptic	boolean
is_jacobian	boolean
is_geometrically_jacobian	boolean
invariants	jsonb	For small genus, a list of geometric invariants (e.g. Igusa). Only possible in the principal case

Table name: curves_fq

Column	Type	Notes
model
???

Things to add for the isogeny class

Column	Type	Notes
size	integer	number of isomorphism classes within the isogeny class
zfv_is_bass	boolean	whether all the over-orders for the order Z[F,V] are Gorenstein
zfv_is_maximal	boolean	whether the order Z[F,V] is maximal
zfv_index	numeric	the index of the order Z[F,V] in the maximal order
zfv_index_factorization	numeric[]	A list of pairs (p, e) giving the factorization of the index
zfv_plus_index	numeric	the index of the order Z[F+V] in the maximal order of the real subfield
zfv_plus_index_factorization	numeric[]	A list of pairs (p, e) giving the factorization of the index
zfv_plus_norm	numeric	The absolute value of the norm of F-V to Z
isogeny_graphs	jsonb	list of pairs (p, G), where p is a degree (or maybe list of degrees) and G is a list of pairs (u,v) representing the directed edge from u to v. Each of u and v is the isom_letter for the corresponding isogeny class
ideal_class_generators	text[]	A list of isom_letters for isomorphism classes that generate the ideal monoid
ideal_class_relations	integer[]	A matrix of positive integers giving relations between the ideal class generators

• Whether or not two isogeny classes come together after base extension, and what the degree is

• Change brauer_invs from a string to list of lists of strings

• Change jsonb types to arrays

• Change nf to text[]: a list of number fields in the non-simple case

• Update galois_t when not set

• Add missing isogeny classes (t^2-p)

• Rename/retype some columns:

• poly (jsonb -> integer[])
• angles (jsonb -> float8[])
• ang_rank -> angle_rank (smallint)
• slps -> slopes (jsonb -> text[])
• A_cnts -> abvar_counts (jsonb -> numeric[])
• A_cnts_str -> abvar_counts_str (text)
• C_cnts -> curve_counts (jsonb -> numeric[])
• C_cnts_str -> curve_counts_str (text)
• pt_cnt -> curve_count (integer)
• is_jac -> has_jacobian (smallint -> boolean)
• is_pp -> has_principal_polarization (smallint -> boolean)
• decomp -> XXX
• is_simp -> is_simple (boolean)
• simple_factors (jsonb -> text[])
• simple_distinct (jsonb -> text[])
• brauer_invs -> brauer_invariants (text -> text[])
• places (jsonb -> text[])
• prim_models -> primitive_models (jsonb -> text[])
• is_prim -> is_primitive (boolean)
• nf -> number_fields (text -> text[])
• galois_t -> galois_groups (smallint -> text[])
• galois_n -> XXX

Unchanged:

• label (text)
• g (smallint)
• q (integer)
• poly_str (text)
• p_rank (smallint)
• dim1_factors (smallint)
• dim2_factors (smallint)
• dim3_factors (smallint)
• dim4_factors (smallint)
• dim5_factors (smallint)
• dim1_distinct (smallint)
• dim2_distinct (smallint)
• dim3_distinct (smallint)
• dim4_distinct (smallint)
• dim5_distinct (smallint)

June Ju and Everett Howe: telling whether an abelian variety is absolutely simple Look at primes dividing discriminant of Weil field, products at most 4g^2. Find pairwise r, hash on multiple r Supersingular if and only if the ultimate field is Q

For Stefano

We'll be doing isogeny classes that Stefano has already computed, plus:

• Any g, q
• ordinary or C-S (q=p, no real roots), squarefree
• Z[F,V] = maximal order

For each isogeny class, write lines to two files

• isomorphism_classes.txt (one line per ideal)

isog_label:frac_ideal:rep_type:is_reduced:cm_elt:is_product
e.g.
1.251.v:{{1,0},{0,1}}:0:f:{21,2}:f

• isogeny_classes.txt (one line per class)

isog_label:order_index:order_is_bass:order_is_maximal:size
e.g.
1.251.v:1:t:t:9

Use \N for null.

["1.251.v", [251,21,1]]