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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2016.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2016.b1 | 2016h3 | \([0, 0, 0, -3036, 64384]\) | \(1036433728/63\) | \(188116992\) | \([2]\) | \(1024\) | \(0.64825\) | |
2016.b2 | 2016h2 | \([0, 0, 0, -1011, -11594]\) | \(306182024/21609\) | \(8065516032\) | \([2]\) | \(1024\) | \(0.64825\) | |
2016.b3 | 2016h1 | \([0, 0, 0, -201, 880]\) | \(19248832/3969\) | \(185177664\) | \([2, 2]\) | \(512\) | \(0.30168\) | \(\Gamma_0(N)\)-optimal |
2016.b4 | 2016h4 | \([0, 0, 0, 429, 5290]\) | \(23393656/45927\) | \(-17142160896\) | \([2]\) | \(1024\) | \(0.64825\) |
Rank
sage: E.rank()
The elliptic curves in class 2016.b have rank \(1\).
Complex multiplication
The elliptic curves in class 2016.b do not have complex multiplication.Modular form 2016.2.a.b
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.