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Database abvar
Description Isogeny classes of abelian varieties over finite fields
Status beta
Contact David Roe
Code abvar/fq
Collections fq_isog

Todo:

  • Compute and add data on Frobenius angle ranks

Collection fq_isog

  • Content: Isogeny classes of abelian varities over finite fields
  • Contributors: Kiran Kedlaya, Taylor Dupuy, David Roe, Christelle Vincent
  • Extent: Complete for pairs (g, q) where q < 500 and q^(g(g+1)/2) < 10^7 (as of October 2016)
Field Description Type of stored data Mathematical type Example of stored data Remarks
_id Mongo id ObjectId - assigned by Mongo; contains creation timestamp
label LMFDB Label string - '2.16.am\_cn' [Labeling Scheme](http://beta.lmfdb.org/Variety/Abelian/Fq/Labels)
g Genus int N 2 The degree of the Weil L-polynomial is 2g.
q Cardinality of Field int prime power 16 All of the roots of the Weil L-polynomial have absolute value 1/q^{1/2}.
poly Coefficients of the Weil L-polynomial string consisting of space separated integers Z^{2g+1} '1 -12 65 -192 256' 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.
angles Frobenius angle numbers list of python floats R^g [0.0826163580681, 0.320878822416] The sorted list (with multiplicity) of $\theta$ with $0 \le \theta \le 1$ and $\frac{1}{\sqrt{q}} e^{\pi i \theta}$ a root of the L-polynomial. There will be g of them unless the list includes 0 or 1.
ang_rank dim(span_Q (1, \theta_1, \dots, \theta_{2g})) - 1, where the roots of the L-polynomial are $1/q^{1/2} e^{\pi i \theta_k}$. int N 3 This might be empty if we haven't computed it yet. Note that the field is plural due to a typo in the original import script.
p_rank The p-rank of the abelian variety int N 2 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.
slps The slopes of the Newton polygon of the Weil polynomial string consisting of space separated rationals Q^{2g+1} '0 1/2 1/2 1' 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).
A_cnts The number of points of the abelian variety over extensions of F_q string consisting of space separated integers N^{10} '1 19 76 171 961 5776 22051 69939 261364 1113799' Counts are given for A(F_{q^n}) for $1 \le n \le max(g,10);$ counts over larger extension fields can be determined from these using the Weil conjectures.
C_cnts The number of points of a corresponding curve over extensions of F_q string consisting of space separated integers Z^{10} '6 9 10 30 87 168 274 513 1086 2178' 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.
pt_cnt The number of points of a corresponding curve int Z 6 If the variety is a Jacobian, this is the point count of a genus g curve of which this is the Jacobian.
is_jac An integer encoding whether the abelian variety is a Jacobian int - 0 1 means that it is definitely a Jacobian, -1 that it is definitely not, and 0 indicates uncertainty.
is_pp An integer encoding whether the abelian variety is principally polarizable int - 0 1 means that it is definitely principally polarizable, -1 that it is definitely not, and 0 indicates uncertainty.
decomp The decomposition into simple factors list of pairs [string, int] - [['2.16.am_cn',1], ['1.16.ah',2]] The first entry in each pair is the label of the factor, the second is its multiplicity.
brauer_invs The Brauer invariants of the endomorphism algebra string consisting of space separated rationals Q^k '0 0 1/2' 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.
places The ideals corresponding to the Brauer invariants of the endomorphism algebra list of lists of strings ((Q^d_i)^e_i)^f [[['0','1'],['1','1/2']],[['0','3']]] 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).
prim_models The isogeny classes over subfields of F_q that yield this class upon base change. list of strings - ['2.2.ab_ab','2.2.b_ab'] If this 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.
nf The label of the number field defined by the Weil polynomial string - '4.0.27792.2' 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.
gal The Galois group of the Weil polynomial bson pair - ('n': 4, 't': 2) If the number field was not in the database when the isogeny class was added to the database or the isogeny class is not simple, 't' will be an empty string.
is_simp Whether this isogeny class is simple bool - True
is_prim Whether this isogeny class is primitive bool - True
sort An order-preserving encoding of g, q, poly string - '209BU' Numbers are encoded in base 62 with digits 0-9, A-Z, a-z. The first character gives g, the second and third give q, and the remainder give the coefficients of the polynomial. The Newton identites and Weil bound are used to find an interval in which each successive coefficient must lie, and then the possibilites are encoded in order.

Index information on collection fq_isog:

  • {'_id': 1} (created by mongo)
  • {'label': 1} (unique)
  • {'sort': 1} (unique)
  • {'g': 1, 'q': 1, 'sort': 1}
  • {'g': 1, 'sort': 1}
  • {'q': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'p_rank': 1, 'sort': 1}
  • {'p_rank': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'ang_rank': 1, 'sort': 1}
  • {'ang_rank': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'pt_cnt': 1, 'sort': 1}
  • {'pt_cnt': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'C_cnts': 1, 'sort': 1}
  • {'C_cnts': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'A_cnts': 1, 'sort': 1}
  • {'A_cnts': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'poly': 1, 'sort': 1}
  • {'poly': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'slps': 1, 'sort': 1}
  • {'slps': 1, 'sort': 1}
  • {'nf': 1, 'sort': 1}
  • {'decomp': 1, 'sort': 1}
  • {'q': 1, 'gal': 1, 'sort': 1}
  • {'gal': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'is_jac': 1, 'sort': 1}
  • {'is_jac': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'is_pp': 1, 'sort': 1}
  • {'is_pp': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'is_simp': 1, 'sort': 1}
  • {'is_simp': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'is_prim': 1, 'sort': 1}
  • {'is_prim': 1, 'sort': 1}
  • {'g': 1, 'q': 1, 'is_simp': 1, 'is_prim': 1, 'sort': 1}