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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 111600.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111600.df1 | 111600eu4 | \([0, 0, 0, -23862675, 44866183250]\) | \(32208729120020809/658986840\) | \(30745690007040000000\) | \([2]\) | \(5308416\) | \(2.8589\) | |
| 111600.df2 | 111600eu2 | \([0, 0, 0, -1542675, 650263250]\) | \(8702409880009/1120910400\) | \(52297195622400000000\) | \([2, 2]\) | \(2654208\) | \(2.5123\) | |
| 111600.df3 | 111600eu1 | \([0, 0, 0, -390675, -83560750]\) | \(141339344329/17141760\) | \(799765954560000000\) | \([2]\) | \(1327104\) | \(2.1658\) | \(\Gamma_0(N)\)-optimal |
| 111600.df4 | 111600eu3 | \([0, 0, 0, 2345325, 3399079250]\) | \(30579142915511/124675335000\) | \(-5816852429760000000000\) | \([2]\) | \(5308416\) | \(2.8589\) |
Rank
sage: E.rank()
The elliptic curves in class 111600.df have rank \(0\).
Complex multiplication
The elliptic curves in class 111600.df do not have complex multiplication.Modular form 111600.2.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.
