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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 2205.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2205.h1 | 2205c2 | \([1, -1, 0, -73509, 5506388]\) | \(55306341/15625\) | \(12410625727828125\) | \([2]\) | \(16128\) | \(1.7950\) | |
| 2205.h2 | 2205c1 | \([1, -1, 0, -27204, -1652365]\) | \(2803221/125\) | \(99285005822625\) | \([2]\) | \(8064\) | \(1.4484\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2205.h have rank \(1\).
Complex multiplication
The elliptic curves in class 2205.h do not have complex multiplication.Modular form 2205.2.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.
