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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 17689.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17689.f1 | 17689g3 | \([0, 1, 1, -13608737, -19327549923]\) | \(-50357871050752/19\) | \(-105163116221611\) | \([]\) | \(408240\) | \(2.4786\) | |
| 17689.f2 | 17689g2 | \([0, 1, 1, -165097, -27524248]\) | \(-89915392/6859\) | \(-37963884956001571\) | \([]\) | \(136080\) | \(1.9293\) | |
| 17689.f3 | 17689g1 | \([0, 1, 1, 11793, -17853]\) | \(32768/19\) | \(-105163116221611\) | \([]\) | \(45360\) | \(1.3800\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 17689.f have rank \(1\).
Complex multiplication
The elliptic curves in class 17689.f do not have complex multiplication.Modular form 17689.2.a.f
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
