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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 162288.fd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162288.fd1 | 162288fr2 | \([0, 0, 0, -431739, -109187190]\) | \(50668941906/1127\) | \(197956440815616\) | \([2]\) | \(786432\) | \(1.8582\) | |
162288.fd2 | 162288fr1 | \([0, 0, 0, -26019, -1833678]\) | \(-22180932/3703\) | \(-325214152768512\) | \([2]\) | \(393216\) | \(1.5116\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 162288.fd have rank \(1\).
Complex multiplication
The elliptic curves in class 162288.fd do not have complex multiplication.Modular form 162288.2.a.fd
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.