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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 169650dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169650.s1 | 169650dr1 | \([1, -1, 0, -78867, -17791709]\) | \(-7620530425/14840982\) | \(-105655037871093750\) | \([]\) | \(1814400\) | \(1.9562\) | \(\Gamma_0(N)\)-optimal |
| 169650.s2 | 169650dr2 | \([1, -1, 0, 680508, 379361416]\) | \(4895482323575/11573848728\) | \(-82395856667109375000\) | \([]\) | \(5443200\) | \(2.5055\) |
Rank
sage: E.rank()
The elliptic curves in class 169650dr have rank \(1\).
Complex multiplication
The elliptic curves in class 169650dr do not have complex multiplication.Modular form 169650.2.a.dr
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 Cremona 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 Cremona labels.
