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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 198550bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
198550.cy1 | 198550bi1 | \([1, 0, 0, -9213, -930583]\) | \(-117649/440\) | \(-323440431875000\) | \([]\) | \(689472\) | \(1.4698\) | \(\Gamma_0(N)\)-optimal |
198550.cy2 | 198550bi2 | \([1, 0, 0, 81037, 22083167]\) | \(80062991/332750\) | \(-244601826605468750\) | \([]\) | \(2068416\) | \(2.0191\) |
Rank
sage: E.rank()
The elliptic curves in class 198550bi have rank \(1\).
Complex multiplication
The elliptic curves in class 198550bi do not have complex multiplication.Modular form 198550.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.