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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1638.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1638.f1 | 1638j3 | \([1, -1, 0, -140967, 20406843]\) | \(-424962187484640625/182\) | \(-132678\) | \([3]\) | \(3240\) | \(1.2298\) | |
| 1638.f2 | 1638j2 | \([1, -1, 0, -1737, 28485]\) | \(-795309684625/6028568\) | \(-4394826072\) | \([3]\) | \(1080\) | \(0.68046\) | |
| 1638.f3 | 1638j1 | \([1, -1, 0, 63, 189]\) | \(37595375/46592\) | \(-33965568\) | \([]\) | \(360\) | \(0.13115\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1638.f have rank \(0\).
Complex multiplication
The elliptic curves in class 1638.f do not have complex multiplication.Modular form 1638.2.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.
