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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 77658r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 77658.r2 | 77658r1 | \([1, 0, 1, 1002119, 465091886]\) | \(9522140375/13502538\) | \(-157820368399918051338\) | \([3]\) | \(2730672\) | \(2.5613\) | \(\Gamma_0(N)\)-optimal |
| 77658.r1 | 77658r2 | \([1, 0, 1, -9731326, -19297327048]\) | \(-8719509765625/8716379112\) | \(-101878784756553957870312\) | \([]\) | \(8192016\) | \(3.1106\) |
Rank
sage: E.rank()
The elliptic curves in class 77658r have rank \(1\).
Complex multiplication
The elliptic curves in class 77658r do not have complex multiplication.Modular form 77658.2.a.r
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.
