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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 491970.z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
491970.z1 | 491970z2 | \([1, 0, 1, -276414, 55923352]\) | \(-15777367606441/3574920\) | \(-529216460303880\) | \([]\) | \(3920400\) | \(1.8174\) | \(\Gamma_0(N)\)-optimal* |
491970.z2 | 491970z1 | \([1, 0, 1, 1311, 267262]\) | \(1685159/209250\) | \(-30976509773250\) | \([]\) | \(1306800\) | \(1.2680\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 491970.z have rank \(0\).
Complex multiplication
The elliptic curves in class 491970.z do not have complex multiplication.Modular form 491970.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.