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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 229320bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 229320.ci4 | 229320bw1 | \([0, 0, 0, 21462, -5207083]\) | \(796706816/8996715\) | \(-12345813556680240\) | \([2]\) | \(1179648\) | \(1.7686\) | \(\Gamma_0(N)\)-optimal |
| 229320.ci3 | 229320bw2 | \([0, 0, 0, -351183, -74593582]\) | \(218156637904/16769025\) | \(368182842163718400\) | \([2, 2]\) | \(2359296\) | \(2.1152\) | |
| 229320.ci2 | 229320bw3 | \([0, 0, 0, -1153803, 389481302]\) | \(1934207124196/373156875\) | \(32772318917869440000\) | \([2]\) | \(4718592\) | \(2.4618\) | |
| 229320.ci1 | 229320bw4 | \([0, 0, 0, -5510883, -4979404402]\) | \(210751929444676/1404585\) | \(123356986434339840\) | \([2]\) | \(4718592\) | \(2.4618\) |
Rank
sage: E.rank()
The elliptic curves in class 229320bw have rank \(1\).
Complex multiplication
The elliptic curves in class 229320bw do not have complex multiplication.Modular form 229320.2.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.
