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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 53040.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53040.cd1 | 53040cl4 | \([0, 1, 0, -22376, 1235124]\) | \(302503589987689/12214946250\) | \(50032419840000\) | \([2]\) | \(196608\) | \(1.3947\) | |
| 53040.cd2 | 53040cl2 | \([0, 1, 0, -3656, -60300]\) | \(1319778683209/395612100\) | \(1620427161600\) | \([2, 2]\) | \(98304\) | \(1.0481\) | |
| 53040.cd3 | 53040cl1 | \([0, 1, 0, -3336, -75276]\) | \(1002702430729/159120\) | \(651755520\) | \([2]\) | \(49152\) | \(0.70158\) | \(\Gamma_0(N)\)-optimal |
| 53040.cd4 | 53040cl3 | \([0, 1, 0, 9944, -392140]\) | \(26546265663191/31856082570\) | \(-130482514206720\) | \([2]\) | \(196608\) | \(1.3947\) |
Rank
sage: E.rank()
The elliptic curves in class 53040.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 53040.cd do not have complex multiplication.Modular form 53040.2.a.cd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
