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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 121520.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121520.cp1 | 121520co2 | \([0, -1, 0, -835760, -293804608]\) | \(133974081659809/192200\) | \(92619316428800\) | \([2]\) | \(884736\) | \(1.9510\) | |
121520.cp2 | 121520co1 | \([0, -1, 0, -51760, -4665408]\) | \(-31824875809/1240000\) | \(-597543976960000\) | \([2]\) | \(442368\) | \(1.6045\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121520.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 121520.cp do not have complex multiplication.Modular form 121520.2.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 LMFDB 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 LMFDB labels.