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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 2200.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2200.f1 | 2200e1 | \([0, 0, 0, -950, -10875]\) | \(379275264/15125\) | \(3781250000\) | \([2]\) | \(1152\) | \(0.60429\) | \(\Gamma_0(N)\)-optimal |
2200.f2 | 2200e2 | \([0, 0, 0, 425, -39750]\) | \(2122416/171875\) | \(-687500000000\) | \([2]\) | \(2304\) | \(0.95087\) |
Rank
sage: E.rank()
The elliptic curves in class 2200.f have rank \(0\).
Complex multiplication
The elliptic curves in class 2200.f do not have complex multiplication.Modular form 2200.2.a.f
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.