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SageMath
E = EllipticCurve("fo1")
E.isogeny_class()
Elliptic curves in class 324870fo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
324870.fo4 | 324870fo1 | \([1, 0, 0, 45814, -2965740]\) | \(90391899763439/84690294000\) | \(-9963728398806000\) | \([2]\) | \(2433024\) | \(1.7570\) | \(\Gamma_0(N)\)-optimal |
324870.fo3 | 324870fo2 | \([1, 0, 0, -237406, -26812864]\) | \(12577973014374481/4642947562500\) | \(546238137780562500\) | \([2, 2]\) | \(4866048\) | \(2.1036\) | |
324870.fo2 | 324870fo3 | \([1, 0, 0, -1645176, 792790830]\) | \(4185743240664514801/113629394531250\) | \(13368384637207031250\) | \([2]\) | \(9732096\) | \(2.4502\) | |
324870.fo1 | 324870fo4 | \([1, 0, 0, -3361156, -2371499614]\) | \(35694515311673154481/10400566692750\) | \(1223616270835344750\) | \([2]\) | \(9732096\) | \(2.4502\) |
Rank
sage: E.rank()
The elliptic curves in class 324870fo have rank \(1\).
Complex multiplication
The elliptic curves in class 324870fo do not have complex multiplication.Modular form 324870.2.a.fo
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.