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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 481650.dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
481650.dz1 | 481650dz2 | \([1, 0, 1, -6838251, 6882167398]\) | \(468898230633769/5540400\) | \(417850821618750000\) | \([2]\) | \(21676032\) | \(2.5312\) | \(\Gamma_0(N)\)-optimal* |
481650.dz2 | 481650dz1 | \([1, 0, 1, -416251, 113379398]\) | \(-105756712489/12476160\) | \(-940938146460000000\) | \([2]\) | \(10838016\) | \(2.1846\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 481650.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 481650.dz do not have complex multiplication.Modular form 481650.2.a.dz
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.