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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 106722bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.w2 | 106722bd1 | \([1, -1, 0, -15606663, 23761644605]\) | \(-35148950502093/46137344\) | \(-551815776965750685696\) | \([2]\) | \(7096320\) | \(2.8868\) | \(\Gamma_0(N)\)-optimal |
| 106722.w1 | 106722bd2 | \([1, -1, 0, -249785223, 1519553778749]\) | \(144106117295241933/247808\) | \(2963854270812137472\) | \([2]\) | \(14192640\) | \(3.2333\) |
Rank
sage: E.rank()
The elliptic curves in class 106722bd have rank \(1\).
Complex multiplication
The elliptic curves in class 106722bd do not have complex multiplication.Modular form 106722.2.a.bd
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
