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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 422142bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 422142.bw3 | 422142bw1 | \([1, 0, 1, -85445, 5617568]\) | \(466025146777/177366672\) | \(26256632968491408\) | \([2]\) | \(3942400\) | \(1.8499\) | \(\Gamma_0(N)\)-optimal |
| 422142.bw2 | 422142bw2 | \([1, 0, 1, -603865, -176658904]\) | \(164503536215257/4178071044\) | \(618504461303698116\) | \([2, 2]\) | \(7884800\) | \(2.1965\) | |
| 422142.bw4 | 422142bw3 | \([1, 0, 1, 99705, -563622404]\) | \(740480746823/927484650666\) | \(-137301014795195752074\) | \([2]\) | \(15769600\) | \(2.5430\) | |
| 422142.bw1 | 422142bw4 | \([1, 0, 1, -9602155, -11453315932]\) | \(661397832743623417/443352042\) | \(65632013677435338\) | \([2]\) | \(15769600\) | \(2.5430\) |
Rank
sage: E.rank()
The elliptic curves in class 422142bw have rank \(0\).
Complex multiplication
The elliptic curves in class 422142bw do not have complex multiplication.Modular form 422142.2.a.bw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
