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SageMath
E = EllipticCurve("hd1")
E.isogeny_class()
Elliptic curves in class 154560.hd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 154560.hd1 | 154560ba4 | \([0, 1, 0, -2944065, -1945308225]\) | \(10765299591712341649/20708625\) | \(5428641792000\) | \([2]\) | \(2162688\) | \(2.1252\) | |
| 154560.hd2 | 154560ba2 | \([0, 1, 0, -184065, -30420225]\) | \(2630872462131649/3645140625\) | \(955551744000000\) | \([2, 2]\) | \(1081344\) | \(1.7786\) | |
| 154560.hd3 | 154560ba3 | \([0, 1, 0, -132545, -47761857]\) | \(-982374577874929/3183837890625\) | \(-834624000000000000\) | \([2]\) | \(2162688\) | \(2.1252\) | |
| 154560.hd4 | 154560ba1 | \([0, 1, 0, -14785, -186817]\) | \(1363569097969/734582625\) | \(192566427648000\) | \([2]\) | \(540672\) | \(1.4321\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 154560.hd have rank \(0\).
Complex multiplication
The elliptic curves in class 154560.hd do not have complex multiplication.Modular form 154560.2.a.hd
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.
