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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 239904.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 239904.df1 | 239904df4 | \([0, 0, 0, -561099, -161770462]\) | \(444893916104/9639\) | \(423270215843328\) | \([2]\) | \(1376256\) | \(1.9230\) | |
| 239904.df2 | 239904df2 | \([0, 0, 0, -148764, 19701920]\) | \(1036433728/122451\) | \(43016795269410816\) | \([2]\) | \(1376256\) | \(1.9230\) | |
| 239904.df3 | 239904df1 | \([0, 0, 0, -36309, -2339260]\) | \(964430272/127449\) | \(699571606741056\) | \([2, 2]\) | \(688128\) | \(1.5764\) | \(\Gamma_0(N)\)-optimal |
| 239904.df4 | 239904df3 | \([0, 0, 0, 56301, -12322618]\) | \(449455096/1753941\) | \(-77019502608824832\) | \([2]\) | \(1376256\) | \(1.9230\) |
Rank
sage: E.rank()
The elliptic curves in class 239904.df have rank \(1\).
Complex multiplication
The elliptic curves in class 239904.df do not have complex multiplication.Modular form 239904.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
