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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 75712.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75712.cd1 | 75712v1 | \([0, 1, 0, -178689, -30643105]\) | \(-226981/14\) | \(-38918682370899968\) | \([]\) | \(599040\) | \(1.9378\) | \(\Gamma_0(N)\)-optimal |
75712.cd2 | 75712v2 | \([0, 1, 0, 524351, 1854769567]\) | \(5735339/537824\) | \(-1495100101960493170688\) | \([]\) | \(2995200\) | \(2.7425\) |
Rank
sage: E.rank()
The elliptic curves in class 75712.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 75712.cd do not have complex multiplication.Modular form 75712.2.a.cd
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.