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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 51520.cl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51520.cl1 | 51520bi4 | \([0, -1, 0, -10677185, -13425075775]\) | \(513516182162686336369/1944885031250\) | \(509839941632000000\) | \([2]\) | \(2875392\) | \(2.6123\) | |
| 51520.cl2 | 51520bi3 | \([0, -1, 0, -677185, -203075775]\) | \(131010595463836369/7704101562500\) | \(2019584000000000000\) | \([2]\) | \(1437696\) | \(2.2658\) | |
| 51520.cl3 | 51520bi2 | \([0, -1, 0, -181825, -3135423]\) | \(2535986675931409/1450751712200\) | \(380305856842956800\) | \([2]\) | \(958464\) | \(2.0630\) | |
| 51520.cl4 | 51520bi1 | \([0, -1, 0, -117825, 15539777]\) | \(690080604747409/3406760000\) | \(893061693440000\) | \([2]\) | \(479232\) | \(1.7165\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 51520.cl have rank \(0\).
Complex multiplication
The elliptic curves in class 51520.cl do not have complex multiplication.Modular form 51520.2.a.cl
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
