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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 429429cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
429429.cn2 | 429429cn1 | \([0, 1, 1, -1239, -5560]\) | \(360448/189\) | \(110384295021\) | \([]\) | \(414720\) | \(0.81065\) | \(\Gamma_0(N)\)-optimal* |
429429.cn1 | 429429cn2 | \([0, 1, 1, -57009, 5220089]\) | \(35084566528/1029\) | \(600981161781\) | \([]\) | \(1244160\) | \(1.3600\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 429429cn have rank \(0\).
Complex multiplication
The elliptic curves in class 429429cn do not have complex multiplication.Modular form 429429.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.