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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 422370.y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
422370.y1 | 422370y4 | \([1, -1, 0, -790701075, -8557690478049]\) | \(1594085333838169257721/5946547230\) | \(203945443387382070270\) | \([2]\) | \(88473600\) | \(3.5359\) | |
422370.y2 | 422370y3 | \([1, -1, 0, -66109095, -35700338925]\) | \(931661646976029241/523087324841250\) | \(17940036843038457924221250\) | \([2]\) | \(88473600\) | \(3.5359\) | \(\Gamma_0(N)\)-optimal* |
422370.y3 | 422370y2 | \([1, -1, 0, -49441725, -133574469039]\) | \(389722452699156121/751636980900\) | \(25778478065834470544100\) | \([2, 2]\) | \(44236800\) | \(3.1893\) | \(\Gamma_0(N)\)-optimal* |
422370.y4 | 422370y1 | \([1, -1, 0, -2071305, -3485821635]\) | \(-28655425171801/136516564080\) | \(-4682033138584448215920\) | \([2]\) | \(22118400\) | \(2.8428\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 422370.y have rank \(0\).
Complex multiplication
The elliptic curves in class 422370.y do not have complex multiplication.Modular form 422370.2.a.y
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.