\\ ec_nfcurves downloaded from the LMFDB on 29 November 2023. \\ Search link: https://www.lmfdb.org/EllipticCurve/ \\ Query "{'field_label': '2.2.56.1', 'jinv': '4956477625/941192,0'}" returned 6 ec_nfcurves, sorted by field. \\ Each entry in the following data list has the form: \\ [Label, Class, Base field, Conductor norm, Rank, Torsion, CM, Sato-Tate, Weierstrass equation] \\ For more details, see the definitions at the bottom of the file. \\ To create a list of ec_nfcurves, type "ec_nfcurves = make_data()" columns = ["short_label", "short_class_label", "field_label", "conductor_norm", "rank", "torsion_structure", "cm", "sato_tate_group", "equation"] data = {[ ["14.1-a4", "14.1-a", "2.2.56.1", 14, 1, , 0, "1.2.A.1.1a", "{y}^2+{x}{y}+{y}={x}^{3}-36{x}-70"], ["14.1-b4", "14.1-b", "2.2.56.1", 14, 0, , 0, "1.2.A.1.1a", "{y}^2+{x}{y}={x}^{3}+\\left(a-1\\right){x}^{2}+\\left(4262a-15944\\right){x}-246560a+922544"], ["98.1-a4", "98.1-a", "2.2.56.1", 98, 1, , 0, "1.2.A.1.1a", "{y}^2+\\left(a+1\\right){x}{y}={x}^{3}+\\left(-a+1\\right){x}^{2}+\\left(993a-3721\\right){x}+29312a-109675"], ["98.1-b4", "98.1-b", "2.2.56.1", 98, 1, , 0, "1.2.A.1.1a", "{y}^2+\\left(a+1\\right){x}{y}={x}^{3}+{x}^{2}+\\left(-993a-3721\\right){x}-29312a-109675"], ["112.1-a4", "112.1-a", "2.2.56.1", 112, 1, , 0, "1.2.A.1.1a", "{y}^2+a{x}{y}+a{y}={x}^{3}-145{x}-702"], ["112.1-f4", "112.1-f", "2.2.56.1", 112, 0, , 0, "1.2.A.1.1a", "{y}^2+a{x}{y}={x}^{3}+\\left(-a+1\\right){x}^{2}+\\left(17047a-63784\\right){x}-1874592a+7014081"] ]} create_record(row) = { out = Map(short_label,row,short_class_label,row,field_label,row,conductor_norm,row,rank,row,torsion_structure,row,cm,row,sato_tate_group,row,equation,row) out } make_data() = { return [create_record(row) | row<-data]; } \\Label (short_label) -- \\ The label of an elliptic curve over a number field $K$ has three components, denoting the conductor, the isogeny class and the isomorphism class: \\ - The conductor label is a label for the conductor, which is an integral ideal of the base field, whose format depends on the field but always includes the norm of the conductor. \\ - The isogeny class label normally consists of one or more letters a-z or A-Z; the ordering of the isogeny classes is based on lexicographical ordering of the Dirichlet coefficients of the L-series. For this to be well-defined, a standard ordering is used for the primes of the base field. In the case of elliptic curves $E$ defined over an imaginary quadratic field $K$, the isogeny class label has a prefix "CM" for curves with complex multiplication by an order in $K$ itself. \\ - The isomorphism class is a positive integer giving the index (starting at 1) of the curve in its isogeny class. \\ Together these give a label of the form $N.a1$ where $N$ is the conductor label, $a$ the class and $1$ the curve number. Omitting the third component gives an isogeny class label, of the form $N.a$. \\ In addition, to specify the elliptic curve completely prepend the label of the base field to give a full label of the curve, for example 3.1.23.1-89.1-A1. \\ Where possible the labelling of isogeny classes matches that of associated cusp forms (Bianchi newforms over imaginary quadratic fields and Hilbert newforms over totally real fields). The prefix "CM" on the isogeny class label for certain elliptic curves over imaginary quadratic fields is used, because the Bianchi modular forms conjecturally attached to such curves are not cuspidal. \\Class (short_class_label) -- \\ The **isogeny class (over a field $K$)** of an elliptic curve $E$ defined over $K$ is the set of all isomorphism classes of elliptic curves defined over $K$ that are isogenous to $E$ over $K$. Over a number field $K$ this is always a finite set; over $\Q$, it has at most 8 elements by a theorem of Kenku \cite{MR0675184, doi:10.1016/0022-314X(82)90025-7}. \\Base field (field_label) -- \\ A **number field** is a finite degree field extension of the field $\Q$ of rational numbers. In LMFDB, number fields are identified by a label. \\ {# Note: I left out the knowl for degree here, since it is technically defining the degree of a number field, rather than a general field extension (which is suggested by this definition)... degree #} \\Conductor norm (conductor_norm) -- \\ The **conductor** of an elliptic curve $E$ defined over a number field $K$ is an ideal of the ring of integers of $K$ that is divisible by the prime ideals of bad reduction and no others. It is defined as \\ $$\\ \mathfrak{n} = \prod_{\mathfrak{p}}\mathfrak{p}^{e_{\mathfrak{p}}} \\$$ \\ where the exponent $e_{\mathfrak{p}}$ is as follows: \\ - $e_{\mathfrak{p}}=0$ if $E$ has good reduction at $\mathfrak{p}$; \\ - $e_{\mathfrak{p}}=1$ if $E$ has multiplicative reduction at $\mathfrak{p}$; \\ - $e_{\mathfrak{p}}=2$ if $E$ has additive reduction at $\mathfrak{p}$ and $\mathfrak{p}$ does not lie above either $2$ or $3$; and \\ - $2\leq e_{\mathfrak{p}}\leq 2+6v_{\mathfrak{p}}(2)+3v_{\mathfrak{p}}(3)$, where $v_{\mathfrak{p}}$ is the valuation at $\mathfrak{p}$, if $E$ has additive reduction and $\mathfrak{p}$ lies above $2$ or $3$. \\ For $\mathfrak{p}=2$ and $3$, there is an algorithm of Tate that simultaneously creates a minimal Weierstrass equation and computes the exponent of the conductor. See: \\
\\
• J. Tate, Algorithm for determining the type of a singular fiber \\ in an elliptic pencil, Modular functions of one variable, IV \\ (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), \\ 33-52. Lecture Notes in Math., Vol. 476, \\ Springer, Berlin, 1975. \\
• J.H. Silverman, Advanced topics in the arithmetic of elliptic \\ curves, GTM 151, Springer-Verlag, New York, 1994. \\
\\ The **conductor norm** is the norm $[\mathcal{O}_K:\mathfrak{n}]$ of the ideal $\mathfrak{n}$. \\ Rank -- \\ The **rank** of an elliptic curve $E$ defined over a number field $K$ is the rank of its Mordell-Weil group $E(K)$. \\ The Mordell-Weil Theorem says that $E(K)$ is a finitely-generated abelian group, hence \\ $E(K) \cong E(K)_{\rm tor} \times \Z^r$ \\ where $E(K)_{\rm tor}$ is the finite torsion subgroup of $E(K)$, and $r\geq 0$ is the **rank**. \\ Rank is an isogeny invariant: all curves in an isogeny class have the same rank. \\Torsion (torsion_structure) -- \\ For an elliptic curve $E$ over a field $K,$ the **torsion subgroup** of $E$ over $K$ is the subgroup $E(K)_{\text{tor}}$ of the Mordell-Weil group $E(K)$ consisting of points of finite order. For a number field $K$ this is always a finite group, since by the Mordell-Weil Theorem $E(K)$ is finitely generated. \\ The torsion subgroup is always either cyclic or a product of two cyclic groups. \\ The **torsion structure** is the list of invariants of the group: \\ - $[]$ for the trivial group; \\ - $[n]$ for a cyclic group of order $n>1$; \\ - $[n_1,n_2]$ with $n_1\mid n_2$ for a product of non-trivial cyclic groups of orders $n_1$ and $n_2$. \\ For $K=\Q$ the possible torsion structures are $[n]$ for $n\le10$ and $n=12$, and $[2,2n]$ for $n=1,2,3,4$. \\ CM -- \\ An elliptic curve whose endomorphism ring is larger than $$\Z$$ is said to have **complex multiplication** (often abbreviated to CM). In this case, for curves defined over fields of characteristic zero, the endomorphism ring is isomorphic to an order in an imaginary quadratic field. The discriminant of this order is the **CM discriminant**. \\ An elliptic curve whose geometric endomorphism ring is larger than $$\Z$$ is said to have **potential complex multiplication** (potential CM). In the literature, these too are often called CM elliptic curves. \\ The property of having potential CM depends only on the $j$-invariant of the curve. In characteristic $0$, CM $j$-invariants are algebraic integers, and there are only finitely many in any given number field. There are precisely 13 CM $j$-invariants in $\Q$ (all integers), associated to the 13 imaginary quadratic orders of class number $1$: \\ $$\\ \begin{array}{c|ccccccccccccc} \\ j & -12288000 & 54000 & 0 & 287496 & 1728 & 16581375 & -3375 & 8000 & -32768 & -884736 & -884736000 & -147197952000 & -262537412640768000\\ \\ \text{CM discriminant} &-27 & -12 & -3 & -16 & -4 & -28 & -7 & -8 & -11 & -19 & -43 & -67 & -163 \\ \end{array} \\$$ \\ CM elliptic curves are examples of CM abelian varieties. \\Sato-Tate (sato_tate_group) -- \\ The **Sato-Tate group** of a motive $X$ is a compact Lie group $G$ containing (as a dense subset) the image of a representation that maps Frobenius elements to conjugacy classes. When $X$ is an Artin motive, $G$ corresponds to the image of the Artin representation; when $X$ is an abelian variety over a number field, one can define $G$ in terms of an $\ell$-adic Galois representation attached to $X$. \\ For motives of even weight $w$ and degree $d$, the Sato-Tate group is a compact subgroup of the orthogonal group $\mathrm{O}(d)$. For motives of odd weight $w$ and even degree $d$, the Sato-Tate group is a compact subgroup of the unitary symplectic group $\mathrm{USp}(d)$. For motives $X$ arising as abelian varieties, the weight is always $w=1$ and the the degree is $d=2g$, where $g$ is the dimension of the variety. \\ The simplest case is when $X$ is an elliptic curve $E/\Q$, in which case $G$ is either $\mathrm{SU}(2)=\mathrm{USp}(2)$ (the generic case), or $G$ is $N(\mathrm{U}(1))$, the normalizer of the subgroup $\mathrm{U}(1)$ of diagonal matrices in $\mathrm{SU}(2)$, which contains $\mathrm{U}(1)$ with index 2. \\ The generalized Sato-Tate conjecture states that when ordered by norm, the sequence of images of Frobenius elements under this representation is equidistributed with respect to the pushforward of the Haar measure of $G$ onto its set of conjugacy classes. \\ This is known for all elliptic curves over totally real number fields (including $\mathbb{Q}$) or CM fields. \\Weierstrass equation (equation) -- \\ A **Weierstrass equation** or **Weierstrass model** over a field $k$ is a plane curve $E$ of the form \\ $$y^2 + a_1xy + a_3y = x^3 + a_2 x^2 + a_4 x + a_6,$$ \\ with $a_1, a_2, a_3, a_4, a_6 \in k$. \\ The **Weierstrass coefficients** of this model $E$ are the five coefficients $a_i$. These are often displayed as a list $[a_1, a_2, a_3, a_4, a_6]$. \\ It is common not to distinguish between the _affine_ curve defined by a Weierstrass equation and its _projective closure_, which contains exactly one additional _point at infinity_, $[0:1:0]$. \\ A Weierstrass model is smooth if and only if its discriminant $\Delta$ is nonzero. In this case, the plane curve $E$ together with the point at infinity as base point, define an elliptic curve defined over $k$. \\ Two smooth Weierstrass models define isomorphic elliptic curves if and only if they are isomorphic as Weierstrass models.