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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 320166.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 320166.cp1 | 320166cp3 | \([1, -1, 0, -56679387, -164228441833]\) | \(-545407363875/14\) | \(-516899591118765162\) | \([]\) | \(22394880\) | \(2.9151\) | |
| 320166.cp2 | 320166cp2 | \([1, -1, 0, -650337, -258306427]\) | \(-7414875/2744\) | \(-11256924428808663528\) | \([]\) | \(7464960\) | \(2.3658\) | |
| 320166.cp3 | 320166cp1 | \([1, -1, 0, 61143, 3565645]\) | \(4492125/3584\) | \(-20168616876452352\) | \([]\) | \(2488320\) | \(1.8165\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 320166.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 320166.cp do not have complex multiplication.Modular form 320166.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
