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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 168259f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 168259.f2 | 168259f1 | \([0, -1, 1, -13559, -604298]\) | \(-43614208/91\) | \(-575244037459\) | \([]\) | \(325080\) | \(1.1421\) | \(\Gamma_0(N)\)-optimal |
| 168259.f3 | 168259f2 | \([0, -1, 1, 23421, -3028337]\) | \(224755712/753571\) | \(-4763595874197979\) | \([]\) | \(975240\) | \(1.6914\) | |
| 168259.f1 | 168259f3 | \([0, -1, 1, -216949, 95931992]\) | \(-178643795968/524596891\) | \(-3316167402387680659\) | \([]\) | \(2925720\) | \(2.2407\) |
Rank
sage: E.rank()
The elliptic curves in class 168259f have rank \(1\).
Complex multiplication
The elliptic curves in class 168259f do not have complex multiplication.Modular form 168259.2.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 Cremona 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 Cremona labels.
