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SageMath
E = EllipticCurve("fz1")
E.isogeny_class()
Elliptic curves in class 84150.fz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
84150.fz1 | 84150ef2 | \([1, -1, 1, -192755, -40030253]\) | \(-2575296504243/765952000\) | \(-235566144000000000\) | \([]\) | \(1036800\) | \(2.0477\) | |
84150.fz2 | 84150ef1 | \([1, -1, 1, 17620, 455247]\) | \(1434104310933/1046272480\) | \(-441396202500000\) | \([]\) | \(345600\) | \(1.4984\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84150.fz have rank \(1\).
Complex multiplication
The elliptic curves in class 84150.fz do not have complex multiplication.Modular form 84150.2.a.fz
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.