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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 45738.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45738.cs1 | 45738cc2 | \([1, -1, 1, -128525, -17702737]\) | \(-545407363875/14\) | \(-6026850522\) | \([]\) | \(155520\) | \(1.3928\) | |
| 45738.cs2 | 45738cc1 | \([1, -1, 1, -1475, -27541]\) | \(-7414875/2744\) | \(-131251411368\) | \([]\) | \(51840\) | \(0.84352\) | \(\Gamma_0(N)\)-optimal |
| 45738.cs3 | 45738cc3 | \([1, -1, 1, 11230, 279073]\) | \(4492125/3584\) | \(-124972772424192\) | \([]\) | \(155520\) | \(1.3928\) |
Rank
sage: E.rank()
The elliptic curves in class 45738.cs have rank \(1\).
Complex multiplication
The elliptic curves in class 45738.cs do not have complex multiplication.Modular form 45738.2.a.cs
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
