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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 338130.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 338130.bd1 | 338130bd4 | \([1, -1, 0, -178415649, -916995163545]\) | \(35694515311673154481/10400566692750\) | \(183011364819123740142750\) | \([2]\) | \(77856768\) | \(3.4431\) | |
| 338130.bd2 | 338130bd3 | \([1, -1, 0, -87328629, 306676883403]\) | \(4185743240664514801/113629394531250\) | \(1999455528825250488281250\) | \([2]\) | \(77856768\) | \(3.4431\) | |
| 338130.bd3 | 338130bd2 | \([1, -1, 0, -12601899, -10358741295]\) | \(12577973014374481/4642947562500\) | \(81698641554701435062500\) | \([2, 2]\) | \(38928384\) | \(3.0965\) | |
| 338130.bd4 | 338130bd1 | \([1, -1, 0, 2431881, -1149047667]\) | \(90391899763439/84690294000\) | \(-1490234787175303494000\) | \([2]\) | \(19464192\) | \(2.7500\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 338130.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 338130.bd do not have complex multiplication.Modular form 338130.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
