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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 227136.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 227136.df1 | 227136iy3 | \([0, -1, 0, -4510497, 3688527777]\) | \(8020417344913/187278\) | \(236966415145893888\) | \([2]\) | \(6193152\) | \(2.4462\) | |
| 227136.df2 | 227136iy2 | \([0, -1, 0, -292257, 53248545]\) | \(2181825073/298116\) | \(377211844517953536\) | \([2, 2]\) | \(3096576\) | \(2.0996\) | |
| 227136.df3 | 227136iy1 | \([0, -1, 0, -75937, -7191263]\) | \(38272753/4368\) | \(5526913472790528\) | \([2]\) | \(1548288\) | \(1.7530\) | \(\Gamma_0(N)\)-optimal |
| 227136.df4 | 227136iy4 | \([0, -1, 0, 464863, 282655905]\) | \(8780064047/32388174\) | \(-40981372536557666304\) | \([2]\) | \(6193152\) | \(2.4462\) |
Rank
sage: E.rank()
The elliptic curves in class 227136.df have rank \(1\).
Complex multiplication
The elliptic curves in class 227136.df do not have complex multiplication.Modular form 227136.2.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.
