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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 54080.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54080.cd1 | 54080bf4 | \([0, 0, 0, -72332, -7487376]\) | \(132304644/5\) | \(1581648773120\) | \([2]\) | \(147456\) | \(1.4268\) | |
| 54080.cd2 | 54080bf2 | \([0, 0, 0, -4732, -105456]\) | \(148176/25\) | \(1977060966400\) | \([2, 2]\) | \(73728\) | \(1.0803\) | |
| 54080.cd3 | 54080bf1 | \([0, 0, 0, -1352, 17576]\) | \(55296/5\) | \(24713262080\) | \([2]\) | \(36864\) | \(0.73369\) | \(\Gamma_0(N)\)-optimal |
| 54080.cd4 | 54080bf3 | \([0, 0, 0, 8788, -597584]\) | \(237276/625\) | \(-197706096640000\) | \([2]\) | \(147456\) | \(1.4268\) |
Rank
sage: E.rank()
The elliptic curves in class 54080.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 54080.cd do not have complex multiplication.Modular form 54080.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
