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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 212160.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.e1 | 212160hc4 | \([0, -1, 0, -15312161, 11979815265]\) | \(12116669613338732028488/5121517181396484375\) | \(167821875000000000000000\) | \([2]\) | \(23756800\) | \(3.1532\) | |
| 212160.e2 | 212160hc2 | \([0, -1, 0, -13164041, 18380783241]\) | \(61593016763229496882624/28164181728515625\) | \(115360488360000000000\) | \([2, 2]\) | \(11878400\) | \(2.8067\) | |
| 212160.e3 | 212160hc1 | \([0, -1, 0, -13162596, 18385020270]\) | \(3940655105874494358404416/824508871875\) | \(52768567800000\) | \([2]\) | \(5939200\) | \(2.4601\) | \(\Gamma_0(N)\)-optimal |
| 212160.e4 | 212160hc3 | \([0, -1, 0, -11039041, 24510558241]\) | \(-4540125474977116360328/5290508867838553125\) | \(-173359394581333708800000\) | \([2]\) | \(23756800\) | \(3.1532\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.e have rank \(0\).
Complex multiplication
The elliptic curves in class 212160.e do not have complex multiplication.Modular form 212160.2.a.e
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.
