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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 722.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 722.e1 | 722e3 | \([1, 1, 1, -30873, 16782247]\) | \(-69173457625/2550136832\) | \(-119973433931988992\) | \([]\) | \(6480\) | \(1.9574\) | |
| 722.e2 | 722e1 | \([1, 1, 1, -5603, -163815]\) | \(-413493625/152\) | \(-7150973912\) | \([]\) | \(720\) | \(0.85878\) | \(\Gamma_0(N)\)-optimal |
| 722.e3 | 722e2 | \([1, 1, 1, 3422, -612177]\) | \(94196375/3511808\) | \(-165216101262848\) | \([]\) | \(2160\) | \(1.4081\) |
Rank
sage: E.rank()
The elliptic curves in class 722.e have rank \(1\).
Complex multiplication
The elliptic curves in class 722.e do not have complex multiplication.Modular form 722.2.a.e
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
