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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 75712df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75712.u1 | 75712df1 | \([0, -1, 0, -178689, 30643105]\) | \(-226981/14\) | \(-38918682370899968\) | \([]\) | \(599040\) | \(1.9378\) | \(\Gamma_0(N)\)-optimal |
| 75712.u2 | 75712df2 | \([0, -1, 0, 524351, -1854769567]\) | \(5735339/537824\) | \(-1495100101960493170688\) | \([]\) | \(2995200\) | \(2.7425\) |
Rank
sage: E.rank()
The elliptic curves in class 75712df have rank \(0\).
Complex multiplication
The elliptic curves in class 75712df do not have complex multiplication.Modular form 75712.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
