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SageMath
sage: E = EllipticCurve("f1")
sage: E.isogeny_class()
Elliptic curves in class 1680.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
1680.f1 | 1680n3 | [0, -1, 0, -1800, 30000] | [4] | 1024 | |
1680.f2 | 1680n2 | [0, -1, 0, -120, 432] | [2, 2] | 512 | |
1680.f3 | 1680n1 | [0, -1, 0, -40, -80] | [2] | 256 | \(\Gamma_0(N)\)-optimal |
1680.f4 | 1680n4 | [0, -1, 0, 280, 2352] | [2] | 1024 |
Rank
sage: E.rank()
The elliptic curves in class 1680.f have rank \(1\).
Complex multiplication
The elliptic curves in class 1680.f do not have complex multiplication.Modular form 1680.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.