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SageMath
sage: E = EllipticCurve("b1")
sage: E.isogeny_class()
Elliptic curves in class 112.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
112.b1 | 112b3 | [0, 0, 0, -299, -1990] | [2] | 16 | |
112.b2 | 112b4 | [0, 0, 0, -59, 138] | [4] | 16 | |
112.b3 | 112b2 | [0, 0, 0, -19, -30] | [2, 2] | 8 | |
112.b4 | 112b1 | [0, 0, 0, 1, -2] | [2] | 4 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 112.b have rank \(0\).
Complex multiplication
The elliptic curves in class 112.b do not have complex multiplication.Modular form 112.2.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.