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SageMath
E = EllipticCurve("dn1")
E.isogeny_class()
Elliptic curves in class 223850.dn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 223850.dn1 | 223850bo3 | \([1, 1, 1, -15956938, 24527577031]\) | \(16232905099479601/4052240\) | \(112168599166250000\) | \([2]\) | \(9953280\) | \(2.6472\) | |
| 223850.dn2 | 223850bo4 | \([1, 1, 1, -15896438, 24722871031]\) | \(-16048965315233521/256572640900\) | \(-7102095066960076562500\) | \([2]\) | \(19906560\) | \(2.9937\) | |
| 223850.dn3 | 223850bo1 | \([1, 1, 1, -226938, 22657031]\) | \(46694890801/18944000\) | \(524382056000000000\) | \([2]\) | \(3317760\) | \(2.0979\) | \(\Gamma_0(N)\)-optimal |
| 223850.dn4 | 223850bo2 | \([1, 1, 1, 741062, 165921031]\) | \(1625964918479/1369000000\) | \(-37894797015625000000\) | \([2]\) | \(6635520\) | \(2.4444\) |
Rank
sage: E.rank()
The elliptic curves in class 223850.dn have rank \(1\).
Complex multiplication
The elliptic curves in class 223850.dn do not have complex multiplication.Modular form 223850.2.a.dn
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
