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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 68400.ex
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68400.ex1 | 68400fg1 | \([0, 0, 0, -207300, 33881375]\) | \(5405726654464/407253125\) | \(74221882031250000\) | \([2]\) | \(552960\) | \(1.9824\) | \(\Gamma_0(N)\)-optimal |
68400.ex2 | 68400fg2 | \([0, 0, 0, 198825, 150439250]\) | \(298091207216/3525390625\) | \(-10280039062500000000\) | \([2]\) | \(1105920\) | \(2.3290\) |
Rank
sage: E.rank()
The elliptic curves in class 68400.ex have rank \(0\).
Complex multiplication
The elliptic curves in class 68400.ex do not have complex multiplication.Modular form 68400.2.a.ex
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.