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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 69678.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69678.h1 | 69678k2 | \([1, -1, 0, -3978, 83754]\) | \(81182737/12482\) | \(1070532722322\) | \([2]\) | \(110592\) | \(1.0318\) | |
| 69678.h2 | 69678k1 | \([1, -1, 0, 432, 7020]\) | \(103823/316\) | \(-27102094236\) | \([2]\) | \(55296\) | \(0.68522\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 69678.h have rank \(1\).
Complex multiplication
The elliptic curves in class 69678.h do not have complex multiplication.Modular form 69678.2.a.h
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
