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SageMath
E = EllipticCurve("cg1")
E.isogeny_class()
Elliptic curves in class 6720.cg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6720.cg1 | 6720y3 | \([0, 1, 0, -23905, 1414655]\) | \(5763259856089/5670\) | \(1486356480\) | \([4]\) | \(12288\) | \(1.0533\) | |
| 6720.cg2 | 6720y2 | \([0, 1, 0, -1505, 21375]\) | \(1439069689/44100\) | \(11560550400\) | \([2, 2]\) | \(6144\) | \(0.70673\) | |
| 6720.cg3 | 6720y1 | \([0, 1, 0, -225, -897]\) | \(4826809/1680\) | \(440401920\) | \([2]\) | \(3072\) | \(0.36016\) | \(\Gamma_0(N)\)-optimal |
| 6720.cg4 | 6720y4 | \([0, 1, 0, 415, 73983]\) | \(30080231/9003750\) | \(-2360279040000\) | \([2]\) | \(12288\) | \(1.0533\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.cg have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.cg do not have complex multiplication.Modular form 6720.2.a.cg
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.
