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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 38720.cz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38720.cz1 | 38720cf2 | \([0, -1, 0, -2018441, -1103078359]\) | \(125330290485184/378125\) | \(2743793676800000\) | \([2]\) | \(460800\) | \(2.1901\) | |
38720.cz2 | 38720cf1 | \([0, -1, 0, -127816, -16725234]\) | \(2036792051776/107421875\) | \(12179481875000000\) | \([2]\) | \(230400\) | \(1.8435\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 38720.cz have rank \(1\).
Complex multiplication
The elliptic curves in class 38720.cz do not have complex multiplication.Modular form 38720.2.a.cz
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.