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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 201810.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
201810.e1 | 201810co2 | \([1, 1, 0, -82743, 5220117]\) | \(2103006527509159/816633498240\) | \(24328328546067840\) | \([2]\) | \(2408448\) | \(1.8431\) | |
201810.e2 | 201810co1 | \([1, 1, 0, 16457, 597397]\) | \(16544338359641/14631321600\) | \(-435881701785600\) | \([2]\) | \(1204224\) | \(1.4965\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 201810.e have rank \(1\).
Complex multiplication
The elliptic curves in class 201810.e do not have complex multiplication.Modular form 201810.2.a.e
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.