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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 86490.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86490.cd1 | 86490bx4 | \([1, -1, 1, -209198, 9679447]\) | \(57960603/31250\) | \(545897967285093750\) | \([2]\) | \(1451520\) | \(2.0945\) | |
86490.cd2 | 86490bx2 | \([1, -1, 1, -122708, -16513569]\) | \(8527173507/200\) | \(4792519877400\) | \([2]\) | \(483840\) | \(1.5452\) | |
86490.cd3 | 86490bx1 | \([1, -1, 1, -7388, -276513]\) | \(-1860867/320\) | \(-7668031803840\) | \([2]\) | \(241920\) | \(1.1986\) | \(\Gamma_0(N)\)-optimal |
86490.cd4 | 86490bx3 | \([1, -1, 1, 50272, 1168831]\) | \(804357/500\) | \(-8734367476561500\) | \([2]\) | \(725760\) | \(1.7479\) |
Rank
sage: E.rank()
The elliptic curves in class 86490.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 86490.cd do not have complex multiplication.Modular form 86490.2.a.cd
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.