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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 289728.cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
289728.cn1 | 289728cn2 | \([0, 0, 0, -18540, 220752]\) | \(3687953625/2024072\) | \(386806182838272\) | \([2]\) | \(774144\) | \(1.4897\) | |
289728.cn2 | 289728cn1 | \([0, 0, 0, 4500, 27216]\) | \(52734375/32192\) | \(-6151987003392\) | \([2]\) | \(387072\) | \(1.1432\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 289728.cn have rank \(2\).
Complex multiplication
The elliptic curves in class 289728.cn do not have complex multiplication.Modular form 289728.2.a.cn
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.