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SageMath
E = EllipticCurve("ds1")
E.isogeny_class()
Elliptic curves in class 68544.ds
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68544.ds1 | 68544dm2 | \([0, 0, 0, -209964, 839792]\) | \(42852953779784/24786408969\) | \(592094564791123968\) | \([2]\) | \(737280\) | \(2.0997\) | |
68544.ds2 | 68544dm1 | \([0, 0, 0, 52476, 104960]\) | \(5352028359488/3098832471\) | \(-9253064177086464\) | \([2]\) | \(368640\) | \(1.7531\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68544.ds have rank \(1\).
Complex multiplication
The elliptic curves in class 68544.ds do not have complex multiplication.Modular form 68544.2.a.ds
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.