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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 414400.cw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414400.cw1 | 414400cw1 | \([0, 0, 0, -1000, -9000]\) | \(6912000/1813\) | \(29008000000\) | \([2]\) | \(258048\) | \(0.71638\) | \(\Gamma_0(N)\)-optimal |
414400.cw2 | 414400cw2 | \([0, 0, 0, 2500, -58000]\) | \(6750000/9583\) | \(-2453248000000\) | \([2]\) | \(516096\) | \(1.0630\) |
Rank
sage: E.rank()
The elliptic curves in class 414400.cw have rank \(0\).
Complex multiplication
The elliptic curves in class 414400.cw do not have complex multiplication.Modular form 414400.2.a.cw
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.