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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 16650bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16650.j2 | 16650bi1 | \([1, -1, 0, -192867, 32155541]\) | \(557238592989/9699328\) | \(13810176000000000\) | \([2]\) | \(138240\) | \(1.8938\) | \(\Gamma_0(N)\)-optimal |
16650.j1 | 16650bi2 | \([1, -1, 0, -3072867, 2074075541]\) | \(2253707317528029/700928\) | \(998001000000000\) | \([2]\) | \(276480\) | \(2.2403\) |
Rank
sage: E.rank()
The elliptic curves in class 16650bi have rank \(1\).
Complex multiplication
The elliptic curves in class 16650bi do not have complex multiplication.Modular form 16650.2.a.bi
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.