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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 22050.cj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.cj1 | 22050bq4 | \([1, -1, 0, -4118067, 3217561591]\) | \(5763259856089/5670\) | \(7598342282343750\) | \([2]\) | \(589824\) | \(2.3406\) | |
| 22050.cj2 | 22050bq2 | \([1, -1, 0, -259317, 49527841]\) | \(1439069689/44100\) | \(59098217751562500\) | \([2, 2]\) | \(294912\) | \(1.9940\) | |
| 22050.cj3 | 22050bq1 | \([1, -1, 0, -38817, -1848659]\) | \(4826809/1680\) | \(2251360676250000\) | \([2]\) | \(147456\) | \(1.6474\) | \(\Gamma_0(N)\)-optimal |
| 22050.cj4 | 22050bq3 | \([1, -1, 0, 71433, 166944091]\) | \(30080231/9003750\) | \(-12065886124277343750\) | \([2]\) | \(589824\) | \(2.3406\) |
Rank
sage: E.rank()
The elliptic curves in class 22050.cj have rank \(1\).
Complex multiplication
The elliptic curves in class 22050.cj do not have complex multiplication.Modular form 22050.2.a.cj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
