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SageMath
E = EllipticCurve("dd1")
E.isogeny_class()
Elliptic curves in class 22050.dd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.dd1 | 22050es4 | \([1, -1, 1, -2951255, -1950620003]\) | \(2121328796049/120050\) | \(160878481657031250\) | \([2]\) | \(589824\) | \(2.3660\) | |
| 22050.dd2 | 22050es3 | \([1, -1, 1, -966755, 342138997]\) | \(74565301329/5468750\) | \(7328648034667968750\) | \([2]\) | \(589824\) | \(2.3660\) | |
| 22050.dd3 | 22050es2 | \([1, -1, 1, -195005, -26757503]\) | \(611960049/122500\) | \(164161715976562500\) | \([2, 2]\) | \(294912\) | \(2.0194\) | |
| 22050.dd4 | 22050es1 | \([1, -1, 1, 25495, -2502503]\) | \(1367631/2800\) | \(-3752267793750000\) | \([2]\) | \(147456\) | \(1.6728\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 22050.dd have rank \(0\).
Complex multiplication
The elliptic curves in class 22050.dd do not have complex multiplication.Modular form 22050.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
