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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 160080.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 160080.cm1 | 160080h4 | \([0, 1, 0, -88600, -10180012]\) | \(18778886261717401/732035835\) | \(2998418780160\) | \([2]\) | \(491520\) | \(1.4785\) | |
| 160080.cm2 | 160080h3 | \([0, 1, 0, -26680, 1534100]\) | \(512787603508921/45649063125\) | \(186978562560000\) | \([4]\) | \(491520\) | \(1.4785\) | |
| 160080.cm3 | 160080h2 | \([0, 1, 0, -5800, -144652]\) | \(5268932332201/900900225\) | \(3690087321600\) | \([2, 2]\) | \(245760\) | \(1.1319\) | |
| 160080.cm4 | 160080h1 | \([0, 1, 0, 680, -12460]\) | \(8477185319/21880935\) | \(-89624309760\) | \([2]\) | \(122880\) | \(0.78531\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 160080.cm have rank \(2\).
Complex multiplication
The elliptic curves in class 160080.cm do not have complex multiplication.Modular form 160080.2.a.cm
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
