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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 4050.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4050.x1 | 4050bb2 | \([1, -1, 1, -5225, 151977]\) | \(-1557792607653/67108864\) | \(-679477248000\) | \([]\) | \(6240\) | \(1.0362\) | |
| 4050.x2 | 4050bb1 | \([1, -1, 1, -50, -123]\) | \(-1339893/4\) | \(-40500\) | \([]\) | \(480\) | \(-0.24627\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4050.x have rank \(1\).
Complex multiplication
The elliptic curves in class 4050.x do not have complex multiplication.Modular form 4050.2.a.x
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.
\(\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
