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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 22050.ew
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.ew1 | 22050fm1 | \([1, -1, 1, -75200, 7913427]\) | \(4386781853/27216\) | \(291776343642000\) | \([2]\) | \(122880\) | \(1.6136\) | \(\Gamma_0(N)\)-optimal |
| 22050.ew2 | 22050fm2 | \([1, -1, 1, -31100, 17086227]\) | \(-310288733/11573604\) | \(-124077890133760500\) | \([2]\) | \(245760\) | \(1.9602\) |
Rank
sage: E.rank()
The elliptic curves in class 22050.ew have rank \(1\).
Complex multiplication
The elliptic curves in class 22050.ew do not have complex multiplication.Modular form 22050.2.a.ew
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
