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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 159936cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
159936.g1 | 159936cp1 | \([0, -1, 0, -587477, 177562701]\) | \(-11632923639808/318495051\) | \(-613918707795542016\) | \([]\) | \(2654208\) | \(2.1945\) | \(\Gamma_0(N)\)-optimal |
159936.g2 | 159936cp2 | \([0, -1, 0, 2611243, 714627789]\) | \(1021544365555712/705905647251\) | \(-1360676347796404617216\) | \([]\) | \(7962624\) | \(2.7438\) |
Rank
sage: E.rank()
The elliptic curves in class 159936cp have rank \(1\).
Complex multiplication
The elliptic curves in class 159936cp do not have complex multiplication.Modular form 159936.2.a.cp
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.