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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 321552.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 321552.ca1 | 321552ca4 | \([0, 0, 0, -179931, -29357750]\) | \(215751695207833/163381911\) | \(487855772135424\) | \([2]\) | \(1703936\) | \(1.7511\) | |
| 321552.ca2 | 321552ca2 | \([0, 0, 0, -13611, -251750]\) | \(93391282153/44876601\) | \(134000812560384\) | \([2, 2]\) | \(851968\) | \(1.4045\) | |
| 321552.ca3 | 321552ca1 | \([0, 0, 0, -7131, 229066]\) | \(13430356633/180873\) | \(540083884032\) | \([2]\) | \(425984\) | \(1.0579\) | \(\Gamma_0(N)\)-optimal |
| 321552.ca4 | 321552ca3 | \([0, 0, 0, 49029, -1917974]\) | \(4365111505607/3058314567\) | \(-9132078364028928\) | \([4]\) | \(1703936\) | \(1.7511\) |
Rank
sage: E.rank()
The elliptic curves in class 321552.ca have rank \(2\).
Complex multiplication
The elliptic curves in class 321552.ca do not have complex multiplication.Modular form 321552.2.a.ca
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.
