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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 66240.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 66240.d1 | 66240er4 | \([0, 0, 0, -17356908, 27821938768]\) | \(3026030815665395929/1364501953125\) | \(260760384000000000000\) | \([2]\) | \(4915200\) | \(2.8753\) | |
| 66240.d2 | 66240er3 | \([0, 0, 0, -9540588, -11145357488]\) | \(502552788401502649/10024505152875\) | \(1915712767641747456000\) | \([2]\) | \(4915200\) | \(2.8753\) | |
| 66240.d3 | 66240er2 | \([0, 0, 0, -1260588, 284354512]\) | \(1159246431432649/488076890625\) | \(93272946315264000000\) | \([2, 2]\) | \(2457600\) | \(2.5287\) | |
| 66240.d4 | 66240er1 | \([0, 0, 0, 262932, 32669008]\) | \(10519294081031/8500170375\) | \(-1624407855169536000\) | \([2]\) | \(1228800\) | \(2.1822\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 66240.d have rank \(1\).
Complex multiplication
The elliptic curves in class 66240.d do not have complex multiplication.Modular form 66240.2.a.d
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.
