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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 6720cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.cj3 | 6720cn1 | \([0, 1, 0, -285, 1683]\) | \(2508888064/118125\) | \(120960000\) | \([2]\) | \(3072\) | \(0.31171\) | \(\Gamma_0(N)\)-optimal |
| 6720.cj2 | 6720cn2 | \([0, 1, 0, -785, -6417]\) | \(3269383504/893025\) | \(14631321600\) | \([2, 2]\) | \(6144\) | \(0.65828\) | |
| 6720.cj1 | 6720cn3 | \([0, 1, 0, -11585, -483777]\) | \(2624033547076/324135\) | \(21242511360\) | \([2]\) | \(12288\) | \(1.0049\) | |
| 6720.cj4 | 6720cn4 | \([0, 1, 0, 2015, -39457]\) | \(13799183324/18600435\) | \(-1218998108160\) | \([2]\) | \(12288\) | \(1.0049\) |
Rank
sage: E.rank()
The elliptic curves in class 6720cn have rank \(1\).
Complex multiplication
The elliptic curves in class 6720cn do not have complex multiplication.Modular form 6720.2.a.cn
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 Cremona 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 Cremona labels.
