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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 123840.dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123840.dm1 | 123840ev1 | \([0, 0, 0, -110028, 12951952]\) | \(770842973809/66873600\) | \(12779743975833600\) | \([2]\) | \(983040\) | \(1.8313\) | \(\Gamma_0(N)\)-optimal |
123840.dm2 | 123840ev2 | \([0, 0, 0, 120372, 60045712]\) | \(1009328859791/8734528080\) | \(-1669194310043566080\) | \([2]\) | \(1966080\) | \(2.1779\) |
Rank
sage: E.rank()
The elliptic curves in class 123840.dm have rank \(1\).
Complex multiplication
The elliptic curves in class 123840.dm do not have complex multiplication.Modular form 123840.2.a.dm
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.