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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 1785.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1785.n1 | 1785j4 | \([1, 0, 1, -13654, 611777]\) | \(281486573281608409/610107421875\) | \(610107421875\) | \([2]\) | \(3072\) | \(1.1451\) | |
| 1785.n2 | 1785j3 | \([1, 0, 1, -11784, -490979]\) | \(180945977944161529/992266372125\) | \(992266372125\) | \([2]\) | \(3072\) | \(1.1451\) | |
| 1785.n3 | 1785j2 | \([1, 0, 1, -1159, 2021]\) | \(171963096231529/97578140625\) | \(97578140625\) | \([2, 2]\) | \(1536\) | \(0.79857\) | |
| 1785.n4 | 1785j1 | \([1, 0, 1, 286, 287]\) | \(2600176603751/1534698375\) | \(-1534698375\) | \([2]\) | \(768\) | \(0.45200\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1785.n have rank \(1\).
Complex multiplication
The elliptic curves in class 1785.n do not have complex multiplication.Modular form 1785.2.a.n
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
