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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 402930.dd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
402930.dd1 | 402930dd3 | \([1, -1, 1, -5744498, -5297956639]\) | \(16232905099479601/4052240\) | \(5233338162700560\) | \([2]\) | \(9953280\) | \(2.3918\) | |
402930.dd2 | 402930dd4 | \([1, -1, 1, -5722718, -5340140143]\) | \(-16048965315233521/256572640900\) | \(-331355347444089332100\) | \([2]\) | \(19906560\) | \(2.7383\) | |
402930.dd3 | 402930dd1 | \([1, -1, 1, -81698, -4893919]\) | \(46694890801/18944000\) | \(24465569204736000\) | \([2]\) | \(3317760\) | \(1.8425\) | \(\Gamma_0(N)\)-optimal* |
402930.dd4 | 402930dd2 | \([1, -1, 1, 266782, -35838943]\) | \(1625964918479/1369000000\) | \(-1768019649561000000\) | \([2]\) | \(6635520\) | \(2.1890\) |
Rank
sage: E.rank()
The elliptic curves in class 402930.dd have rank \(1\).
Complex multiplication
The elliptic curves in class 402930.dd do not have complex multiplication.Modular form 402930.2.a.dd
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.