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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 49419f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 49419.j3 | 49419f1 | \([0, 0, 1, 1734, 1228]\) | \(32768/19\) | \(-334329468219\) | \([]\) | \(40320\) | \(0.90074\) | \(\Gamma_0(N)\)-optimal |
| 49419.j2 | 49419f2 | \([0, 0, 1, -24276, 1548823]\) | \(-89915392/6859\) | \(-120692938027059\) | \([]\) | \(120960\) | \(1.4500\) | |
| 49419.j1 | 49419f3 | \([0, 0, 1, -2001036, 1089508108]\) | \(-50357871050752/19\) | \(-334329468219\) | \([]\) | \(362880\) | \(1.9994\) |
Rank
sage: E.rank()
The elliptic curves in class 49419f have rank \(1\).
Complex multiplication
The elliptic curves in class 49419f do not have complex multiplication.Modular form 49419.2.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.
