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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 405042.ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405042.ca1 | 405042ca2 | \([1, 1, 1, -442774, 65302091]\) | \(204055591784617/78708537864\) | \(3702912506033738184\) | \([2]\) | \(8805888\) | \(2.2620\) | \(\Gamma_0(N)\)-optimal* |
405042.ca2 | 405042ca1 | \([1, 1, 1, -197294, -33086293]\) | \(18052771191337/444958272\) | \(20933453914477632\) | \([2]\) | \(4402944\) | \(1.9154\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 405042.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 405042.ca do not have complex multiplication.Modular form 405042.2.a.ca
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.