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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 486720.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.cp1 | 486720cp2 | \([0, 0, 0, -82987788, -290983299568]\) | \(68523370149961/243360\) | \(224479538492725002240\) | \([2]\) | \(41287680\) | \(3.1233\) | \(\Gamma_0(N)\)-optimal* |
486720.cp2 | 486720cp1 | \([0, 0, 0, -5112588, -4682914288]\) | \(-16022066761/998400\) | \(-920941696380410265600\) | \([2]\) | \(20643840\) | \(2.7768\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 486720.cp do not have complex multiplication.Modular form 486720.2.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.