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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 258570eh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.eh4 | 258570eh1 | \([1, -1, 1, 1422103, -513915879]\) | \(90391899763439/84690294000\) | \(-298003443629755734000\) | \([2]\) | \(11354112\) | \(2.6158\) | \(\Gamma_0(N)\)-optimal |
| 258570.eh3 | 258570eh2 | \([1, -1, 1, -7369277, -4631798271]\) | \(12577973014374481/4642947562500\) | \(16337342768197032562500\) | \([2, 2]\) | \(22708224\) | \(2.9624\) | |
| 258570.eh2 | 258570eh3 | \([1, -1, 1, -51067607, 137143063581]\) | \(4185743240664514801/113629394531250\) | \(399832723073043457031250\) | \([2]\) | \(45416448\) | \(3.3090\) | |
| 258570.eh1 | 258570eh4 | \([1, -1, 1, -104333027, -410056629771]\) | \(35694515311673154481/10400566692750\) | \(36596929160978466432750\) | \([2]\) | \(45416448\) | \(3.3090\) |
Rank
sage: E.rank()
The elliptic curves in class 258570eh have rank \(0\).
Complex multiplication
The elliptic curves in class 258570eh do not have complex multiplication.Modular form 258570.2.a.eh
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.
