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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 16650.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16650.b1 | 16650bd3 | \([1, -1, 0, -72103167, 48533356741]\) | \(3639478711331685826729/2016912141902025000\) | \(22973889866352753515625000\) | \([2]\) | \(4423680\) | \(3.5568\) | |
| 16650.b2 | 16650bd2 | \([1, -1, 0, -43978167, -111582268259]\) | \(825824067562227826729/5613755625000000\) | \(63944185166015625000000\) | \([2, 2]\) | \(2211840\) | \(3.2102\) | |
| 16650.b3 | 16650bd1 | \([1, -1, 0, -43906167, -111967972259]\) | \(821774646379511057449/38361600000\) | \(436962600000000000\) | \([2]\) | \(1105920\) | \(2.8637\) | \(\Gamma_0(N)\)-optimal |
| 16650.b4 | 16650bd4 | \([1, -1, 0, -17005167, -247013701259]\) | \(-47744008200656797609/2286529541015625000\) | \(-26045000553131103515625000\) | \([2]\) | \(4423680\) | \(3.5568\) |
Rank
sage: E.rank()
The elliptic curves in class 16650.b have rank \(1\).
Complex multiplication
The elliptic curves in class 16650.b do not have complex multiplication.Modular form 16650.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
