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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 21780.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21780.bd1 | 21780bd4 | \([0, 0, 0, -1694847, -461332586]\) | \(1628514404944/664335375\) | \(219639771516442464000\) | \([2]\) | \(829440\) | \(2.6009\) | |
| 21780.bd2 | 21780bd2 | \([0, 0, 0, -780087, 265169806]\) | \(158792223184/16335\) | \(5400609094045440\) | \([2]\) | \(276480\) | \(2.0516\) | |
| 21780.bd3 | 21780bd1 | \([0, 0, 0, -45012, 4806241]\) | \(-488095744/200475\) | \(-4142512657364400\) | \([2]\) | \(138240\) | \(1.7050\) | \(\Gamma_0(N)\)-optimal |
| 21780.bd4 | 21780bd3 | \([0, 0, 0, 347028, -52549211]\) | \(223673040896/187171875\) | \(-3867623700162750000\) | \([2]\) | \(414720\) | \(2.2543\) |
Rank
sage: E.rank()
The elliptic curves in class 21780.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 21780.bd do not have complex multiplication.Modular form 21780.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
