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SageMath
E = EllipticCurve("ee1")
E.isogeny_class()
Elliptic curves in class 68400ee
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68400.cd2 | 68400ee1 | \([0, 0, 0, -55875, 5085250]\) | \(-413493625/152\) | \(-7091712000000\) | \([]\) | \(207360\) | \(1.4337\) | \(\Gamma_0(N)\)-optimal |
| 68400.cd3 | 68400ee2 | \([0, 0, 0, 34125, 19323250]\) | \(94196375/3511808\) | \(-163846914048000000\) | \([]\) | \(622080\) | \(1.9830\) | |
| 68400.cd1 | 68400ee3 | \([0, 0, 0, -307875, -528902750]\) | \(-69173457625/2550136832\) | \(-118979184033792000000\) | \([]\) | \(1866240\) | \(2.5323\) |
Rank
sage: E.rank()
The elliptic curves in class 68400ee have rank \(1\).
Complex multiplication
The elliptic curves in class 68400ee do not have complex multiplication.Modular form 68400.2.a.ee
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.
