# Abelian variety isogeny classes downloaded from the LMFDB on 08 April 2026. # Search link: https://www.lmfdb.org/Variety/Abelian/Fq/1/71/ # Query "{'q': 71, 'g': 1}" returned 33 classes, sorted by dimension. # Each entry in the following data list has the form: # [Label, Dimension, Base field, L-polynomial, $p$-rank, Isogeny factors] # For more details, see the definitions at the bottom of the file. "1.71.aq" 1 71 [1, -16, 71] 1 [["1.71.aq", 1]] "1.71.ap" 1 71 [1, -15, 71] 1 [["1.71.ap", 1]] "1.71.ao" 1 71 [1, -14, 71] 1 [["1.71.ao", 1]] "1.71.an" 1 71 [1, -13, 71] 1 [["1.71.an", 1]] "1.71.am" 1 71 [1, -12, 71] 1 [["1.71.am", 1]] "1.71.al" 1 71 [1, -11, 71] 1 [["1.71.al", 1]] "1.71.ak" 1 71 [1, -10, 71] 1 [["1.71.ak", 1]] "1.71.aj" 1 71 [1, -9, 71] 1 [["1.71.aj", 1]] "1.71.ai" 1 71 [1, -8, 71] 1 [["1.71.ai", 1]] "1.71.ah" 1 71 [1, -7, 71] 1 [["1.71.ah", 1]] "1.71.ag" 1 71 [1, -6, 71] 1 [["1.71.ag", 1]] "1.71.af" 1 71 [1, -5, 71] 1 [["1.71.af", 1]] "1.71.ae" 1 71 [1, -4, 71] 1 [["1.71.ae", 1]] "1.71.ad" 1 71 [1, -3, 71] 1 [["1.71.ad", 1]] "1.71.ac" 1 71 [1, -2, 71] 1 [["1.71.ac", 1]] "1.71.ab" 1 71 [1, -1, 71] 1 [["1.71.ab", 1]] "1.71.a" 1 71 [1, 0, 71] 0 [["1.71.a", 1]] "1.71.b" 1 71 [1, 1, 71] 1 [["1.71.b", 1]] "1.71.c" 1 71 [1, 2, 71] 1 [["1.71.c", 1]] "1.71.d" 1 71 [1, 3, 71] 1 [["1.71.d", 1]] "1.71.e" 1 71 [1, 4, 71] 1 [["1.71.e", 1]] "1.71.f" 1 71 [1, 5, 71] 1 [["1.71.f", 1]] "1.71.g" 1 71 [1, 6, 71] 1 [["1.71.g", 1]] "1.71.h" 1 71 [1, 7, 71] 1 [["1.71.h", 1]] "1.71.i" 1 71 [1, 8, 71] 1 [["1.71.i", 1]] "1.71.j" 1 71 [1, 9, 71] 1 [["1.71.j", 1]] "1.71.k" 1 71 [1, 10, 71] 1 [["1.71.k", 1]] "1.71.l" 1 71 [1, 11, 71] 1 [["1.71.l", 1]] "1.71.m" 1 71 [1, 12, 71] 1 [["1.71.m", 1]] "1.71.n" 1 71 [1, 13, 71] 1 [["1.71.n", 1]] "1.71.o" 1 71 [1, 14, 71] 1 [["1.71.o", 1]] "1.71.p" 1 71 [1, 15, 71] 1 [["1.71.p", 1]] "1.71.q" 1 71 [1, 16, 71] 1 [["1.71.q", 1]] # Label -- # The LMFDB uses a systematic system to label # isogeny classes, # endomorphism ring, # weak equivalence classes, # Deligne module, and # polarizations defined over finite fields. # The **label** format for an isogeny class defined over a finite field is **g.q.isog**, where # - **$g$** is the dimension of the abelian varieties contained in the isogeny class, # - **$q$** is the cardinality of the field over which the abelian varieties and the isogenies are defined, and # - **isog** specifies the isogeny class. # The label **isog** is obtained in the following manner: If the Weil $q$-polynomial of the isogeny class is # $$1 + a_1 x + a_2 x^2 + \cdots +a_gx^g+ qa_{g-1}x^{g+1} \cdots + q^{g-1}a_1 x^{2g-1} + q^g x^{2g},$$ # the label contains the integer coefficients $a_1, \ldots a_g$, encoded in base 26 with the symbols a, b, c... z, where a = 0, and separated by underscores. Negative numbers are distinguished from positive numbers by a leading a. For example, ae_j_ap denotes the polynomial # $$1 -4x + 9x^2 - 15x^3 +9qx^4 - 4q^2 x^5 + q^3 x^6,$$ # where $q$ is the cardinality of the field, because e = 4, j = 9 and p = 15. # Each endomorphism ring of an ordinary abelian variety has a label of the form **g.q.isog.N.i**, where # - **$N :=[\mathcal{O}_{\mathbb{Q}[F]}:R]$** is the index of the endomorphism ring $R$ in the maximal order $\mathcal{O}_{\mathbb{Q}[F]}$ of the field generated by the Frobenius endomorphism $\mathbb{Q}[F]$, # - **$i$** is an index that uniquely determines the endomorphism ring among all the overorders of $\Z[F,V]$ with index $N$ in $\mathcal{O}_{\mathbb{Q}[F]}$. # Each weak equivalence class of an ordinary isogeny class has a label of the form **g.q.isog.N.i.w**, where # - **$w$** is an index that uniquely determines the weak equivalence class class among all the with the same endomorphism ring. # Each Deligne module of an ordinary isogeny class has a label of the form **g.q.isog.N.i.w.j**, where # - **$j$** is an index that uniquely determines the isomorphism class of the Deligne module representing the unpolarized abelian variety within the same weak equivalence class. # Each polarizations of an ordinary isogeny class has a label of the form **g.q.isog.N.i.w.j.d.k**, where # - **$d$** is the degree of the polarization, # - **$k$** is an index that determines the polarization among all the ones with the same degree. #Dimension (g) -- # The **dimension** of an algebraic variety $V$ is the maximal length $d$ of a chain # $$ # V_0 \subset V_1 \subset \cdots \subset V_d # $$ # of distinct irreducible subvarieties of $V$. #Base field (q) -- # The **base field**, of an algebraic variety is the field over which it is defined; it necessarily contains the coefficients of a set of defining equations for the variety, but it is not necessarily a minimal field of definition. #L-polynomial (polynomial) -- # Let $A$ be an abelian variety of dimension $g$ defined over $\F_q$. Let $F_q$ be the inverse of the field automorphism $x \mapsto x^q$ in $\Gal(\overline{\F}_q/\F_q)$, which acts on $\ell$-adic \'etale cohomology. The **L-polynomial** of $A$ is # $$L_A(t) = \det(1-t F_q|H^1(A_{\overline{\F}_q}, \Q_\ell)).$$ # This is a polynomial of degree $2g$ with integer coefficients that are independent of $\ell$. Its constant term is $1$. # The L-polynomial $L_A(t)$ is the reverse of the characteristic polynomial $P_A(t)$, which is a Weil $q$-polynomial. Thus the complex roots of $L_A(t)$ have absolute value $q^{-1/2}$. #$p$-rank (p_rank) -- # Let $A$ be a $g$-dimensional abelian variety over $\F_q$ where $q=p^r$. # The **$p$-rank** of $A$ is the dimension of the geometric $p$-torsion as an $\F_p$-vector space: $$p\operatorname{-rank}(A) = \dim_{\F_p}( A(\overline{\F}_p)[p] ).$$ The $p$-rank is at most $g$, with equality if and only if $A$ is ordinary. The difference between $g$ and the $p$-rank is the **$p$-corank** of $A$. #Isogeny factors (decompositionraw) -- # Any abelian variety $A$ is isogenous to a product of simple abelian varieties $B_i$, called the **isogeny factors** of $A$: # $$A \sim B_1 \times \cdots \times B_n.$$ # We say that $A$ decomposes up to isogeny into the product of the $B_i$. # Note that two elements of this product might be isogenous; in other words, elements of the decomposition may appear with multiplicity.