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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 261450.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
261450.i1 | 261450i2 | \([1, -1, 0, -682542, -204405134]\) | \(3087199234101529/199326394890\) | \(2270452216793906250\) | \([2]\) | \(5898240\) | \(2.2725\) | |
261450.i2 | 261450i1 | \([1, -1, 0, -131292, 14441116]\) | \(21973174804729/4842576900\) | \(55159977501562500\) | \([2]\) | \(2949120\) | \(1.9259\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 261450.i have rank \(2\).
Complex multiplication
The elliptic curves in class 261450.i do not have complex multiplication.Modular form 261450.2.a.i
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.