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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 139230.cn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139230.cn1 | 139230bo2 | \([1, -1, 1, -1405868, -641229969]\) | \(421531012285745314681/14601840926400\) | \(10644742035345600\) | \([2]\) | \(2211840\) | \(2.1661\) | |
139230.cn2 | 139230bo1 | \([1, -1, 1, -83948, -10938513]\) | \(-89747507348586361/19239456337920\) | \(-14025563670343680\) | \([2]\) | \(1105920\) | \(1.8195\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 139230.cn have rank \(1\).
Complex multiplication
The elliptic curves in class 139230.cn do not have complex multiplication.Modular form 139230.2.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.