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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 5610.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5610.bf1 | 5610bj4 | \([1, 0, 0, -84486, 9444996]\) | \(66692696957462376289/1322972640\) | \(1322972640\) | \([2]\) | \(20480\) | \(1.2827\) | |
| 5610.bf2 | 5610bj3 | \([1, 0, 0, -8006, -21180]\) | \(56751044592329569/32660264340000\) | \(32660264340000\) | \([2]\) | \(20480\) | \(1.2827\) | |
| 5610.bf3 | 5610bj2 | \([1, 0, 0, -5286, 146916]\) | \(16334668434139489/72511718400\) | \(72511718400\) | \([2, 2]\) | \(10240\) | \(0.93612\) | |
| 5610.bf4 | 5610bj1 | \([1, 0, 0, -166, 4580]\) | \(-506071034209/8823767040\) | \(-8823767040\) | \([2]\) | \(5120\) | \(0.58954\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5610.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 5610.bf do not have complex multiplication.Modular form 5610.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
