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SageMath
E = EllipticCurve("ki1")
E.isogeny_class()
Elliptic curves in class 388080.ki
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.ki1 | 388080ki4 | \([0, 0, 0, -7244307, -7504873614]\) | \(239369344910082/385\) | \(67624871086080\) | \([2]\) | \(6291456\) | \(2.3470\) | |
| 388080.ki2 | 388080ki3 | \([0, 0, 0, -576387, -48212766]\) | \(120564797922/64054375\) | \(11251087926946560000\) | \([2]\) | \(6291456\) | \(2.3470\) | |
| 388080.ki3 | 388080ki2 | \([0, 0, 0, -452907, -117188694]\) | \(116986321764/148225\) | \(13017787684070400\) | \([2, 2]\) | \(3145728\) | \(2.0004\) | |
| 388080.ki4 | 388080ki1 | \([0, 0, 0, -20727, -2833866]\) | \(-44851536/132055\) | \(-2899416347815680\) | \([2]\) | \(1572864\) | \(1.6539\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 388080.ki have rank \(0\).
Complex multiplication
The elliptic curves in class 388080.ki do not have complex multiplication.Modular form 388080.2.a.ki
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
