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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 41971.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 41971.a1 | 41971a3 | \([0, 1, 1, -1699457, 852167382]\) | \(-50357871050752/19\) | \(-204805091251\) | \([]\) | \(316710\) | \(1.9585\) | |
| 41971.a2 | 41971a2 | \([0, 1, 1, -20617, 1205357]\) | \(-89915392/6859\) | \(-73934637941611\) | \([]\) | \(105570\) | \(1.4092\) | |
| 41971.a3 | 41971a1 | \([0, 1, 1, 1473, 1452]\) | \(32768/19\) | \(-204805091251\) | \([]\) | \(35190\) | \(0.85990\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 41971.a have rank \(0\).
Complex multiplication
The elliptic curves in class 41971.a do not have complex multiplication.Modular form 41971.2.a.a
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.
