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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 20160fd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20160.em1 | 20160fd1 | \([0, 0, 0, -192, 376]\) | \(1048576/525\) | \(391910400\) | \([2]\) | \(6144\) | \(0.34167\) | \(\Gamma_0(N)\)-optimal |
20160.em2 | 20160fd2 | \([0, 0, 0, 708, 2896]\) | \(3286064/2205\) | \(-26336378880\) | \([2]\) | \(12288\) | \(0.68824\) |
Rank
sage: E.rank()
The elliptic curves in class 20160fd have rank \(1\).
Complex multiplication
The elliptic curves in class 20160fd do not have complex multiplication.Modular form 20160.2.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 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.