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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 8400.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8400.cd1 | 8400cp2 | \([0, 1, 0, -1443708, 666974088]\) | \(665567485783184/257298363\) | \(128649181500000000\) | \([2]\) | \(161280\) | \(2.2484\) | |
8400.cd2 | 8400cp1 | \([0, 1, 0, -76833, 13607838]\) | \(-1605176213504/1640558367\) | \(-51267448968750000\) | \([2]\) | \(80640\) | \(1.9018\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8400.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 8400.cd do not have complex multiplication.Modular form 8400.2.a.cd
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.