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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 72450.cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.cf1 | 72450cj1 | \([1, -1, 0, -24687, -1338579]\) | \(18260010268037/1994194944\) | \(181721014272000\) | \([2]\) | \(294912\) | \(1.4698\) | \(\Gamma_0(N)\)-optimal |
72450.cf2 | 72450cj2 | \([1, -1, 0, 32913, -6695379]\) | \(43269428370043/237036554496\) | \(-21599956028448000\) | \([2]\) | \(589824\) | \(1.8164\) |
Rank
sage: E.rank()
The elliptic curves in class 72450.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.cf do not have complex multiplication.Modular form 72450.2.a.cf
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.