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SageMath
E = EllipticCurve("dw1")
E.isogeny_class()
Elliptic curves in class 139650.dw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139650.dw1 | 139650gy4 | \([1, 0, 1, -64708720151, 6335661403055198]\) | \(16300610738133468173382620881/2228489100\) | \(4096554908217187500\) | \([2]\) | \(207360000\) | \(4.3865\) | |
| 139650.dw2 | 139650gy3 | \([1, 0, 1, -4044294651, 98994475102198]\) | \(-3979640234041473454886161/1471455901872240\) | \(-2704926803115111933750000\) | \([2]\) | \(103680000\) | \(4.0399\) | |
| 139650.dw3 | 139650gy2 | \([1, 0, 1, -107732651, 370801730198]\) | \(75224183150104868881/11219310000000000\) | \(20624071909218750000000000\) | \([2]\) | \(41472000\) | \(3.5818\) | |
| 139650.dw4 | 139650gy1 | \([1, 0, 1, 11435349, 31649602198]\) | \(89962967236397039/287450726400000\) | \(-528410789222400000000000\) | \([2]\) | \(20736000\) | \(3.2352\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 139650.dw have rank \(0\).
Complex multiplication
The elliptic curves in class 139650.dw do not have complex multiplication.Modular form 139650.2.a.dw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
