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SageMath
E = EllipticCurve("hm1")
E.isogeny_class()
Elliptic curves in class 235950hm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
235950.hm2 | 235950hm1 | \([1, 0, 0, -5523713, 5063991417]\) | \(-673350049820449/10617750000\) | \(-293906121996093750000\) | \([2]\) | \(13271040\) | \(2.7289\) | \(\Gamma_0(N)\)-optimal |
235950.hm1 | 235950hm2 | \([1, 0, 0, -88711213, 321592428917]\) | \(2789222297765780449/677605500\) | \(18756554331023437500\) | \([2]\) | \(26542080\) | \(3.0754\) |
Rank
sage: E.rank()
The elliptic curves in class 235950hm have rank \(1\).
Complex multiplication
The elliptic curves in class 235950hm do not have complex multiplication.Modular form 235950.2.a.hm
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.