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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 335160.cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 335160.cz1 | 335160cz3 | \([0, 0, 0, -13406547, 18893982814]\) | \(3034301922374404/1425\) | \(125149923763200\) | \([2]\) | \(6291456\) | \(2.4787\) | |
| 335160.cz2 | 335160cz4 | \([0, 0, 0, -1005627, 168660646]\) | \(1280615525284/601171875\) | \(52797624087600000000\) | \([2]\) | \(6291456\) | \(2.4787\) | |
| 335160.cz3 | 335160cz2 | \([0, 0, 0, -838047, 295116514]\) | \(2964647793616/2030625\) | \(44584660340640000\) | \([2, 2]\) | \(3145728\) | \(2.1321\) | |
| 335160.cz4 | 335160cz1 | \([0, 0, 0, -42042, 6485101]\) | \(-5988775936/9774075\) | \(-13412551985809200\) | \([2]\) | \(1572864\) | \(1.7856\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 335160.cz have rank \(1\).
Complex multiplication
The elliptic curves in class 335160.cz do not have complex multiplication.Modular form 335160.2.a.cz
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.
