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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 119952.fd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
119952.fd1 | 119952gn2 | \([0, 0, 0, -772779, -261012710]\) | \(423564751/867\) | \(104469359939997696\) | \([2]\) | \(1835008\) | \(2.1520\) | |
119952.fd2 | 119952gn1 | \([0, 0, 0, -31899, -6890870]\) | \(-29791/153\) | \(-18435769401176064\) | \([2]\) | \(917504\) | \(1.8054\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 119952.fd have rank \(1\).
Complex multiplication
The elliptic curves in class 119952.fd do not have complex multiplication.Modular form 119952.2.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.