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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 42042.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42042.h1 | 42042g2 | \([1, 1, 0, -81120, 8225568]\) | \(501796540869625/39902230368\) | \(4694457500564832\) | \([2]\) | \(245760\) | \(1.7506\) | |
42042.h2 | 42042g1 | \([1, 1, 0, 5120, 584704]\) | \(126128378375/1319205888\) | \(-155203253517312\) | \([2]\) | \(122880\) | \(1.4040\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42042.h have rank \(2\).
Complex multiplication
The elliptic curves in class 42042.h do not have complex multiplication.Modular form 42042.2.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.