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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 364560.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 364560.df1 | 364560df3 | \([0, -1, 0, -5196760, 4561465840]\) | \(32208729120020809/658986840\) | \(317559368659599360\) | \([2]\) | \(10616832\) | \(2.4778\) | |
| 364560.df2 | 364560df2 | \([0, -1, 0, -335960, 66198000]\) | \(8702409880009/1120910400\) | \(540155853412761600\) | \([2, 2]\) | \(5308416\) | \(2.1313\) | |
| 364560.df3 | 364560df1 | \([0, -1, 0, -85080, -8463888]\) | \(141339344329/17141760\) | \(8260447937495040\) | \([2]\) | \(2654208\) | \(1.7847\) | \(\Gamma_0(N)\)-optimal |
| 364560.df4 | 364560df4 | \([0, -1, 0, 510760, 345276912]\) | \(30579142915511/124675335000\) | \(-60079835084451840000\) | \([2]\) | \(10616832\) | \(2.4778\) |
Rank
sage: E.rank()
The elliptic curves in class 364560.df have rank \(0\).
Complex multiplication
The elliptic curves in class 364560.df do not have complex multiplication.Modular form 364560.2.a.df
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
