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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 68544cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68544.o3 | 68544cp1 | \([0, 0, 0, 62196, -161118736]\) | \(139233463487/58763045376\) | \(-11229792850176638976\) | \([]\) | \(1658880\) | \(2.3345\) | \(\Gamma_0(N)\)-optimal |
| 68544.o2 | 68544cp2 | \([0, 0, 0, -559884, 4355430896]\) | \(-101566487155393/42823570577256\) | \(-8183711780259659513856\) | \([]\) | \(4976640\) | \(2.8838\) | |
| 68544.o1 | 68544cp3 | \([0, 0, 0, -219860364, 1254825734384]\) | \(-6150311179917589675873/244053849830826\) | \(-46639417006927945138176\) | \([]\) | \(14929920\) | \(3.4331\) |
Rank
sage: E.rank()
The elliptic curves in class 68544cp have rank \(2\).
Complex multiplication
The elliptic curves in class 68544cp do not have complex multiplication.Modular form 68544.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
