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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 162450n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162450.cq4 | 162450n1 | \([1, -1, 1, 298622020, -11920950932353]\) | \(5495662324535111/117739817533440\) | \(-63094647517851722711040000000\) | \([2]\) | \(154828800\) | \(4.2065\) | \(\Gamma_0(N)\)-optimal |
162450.cq3 | 162450n2 | \([1, -1, 1, -6355329980, -184710776468353]\) | \(52974743974734147769/3152005008998400\) | \(1689102711245271811334400000000\) | \([2, 2]\) | \(309657600\) | \(4.5531\) | |
162450.cq2 | 162450n3 | \([1, -1, 1, -18987441980, 777552987243647]\) | \(1412712966892699019449/330160465517040000\) | \(176927046707975716728483750000000\) | \([2]\) | \(619315200\) | \(4.8997\) | |
162450.cq1 | 162450n4 | \([1, -1, 1, -100186449980, -12205603221908353]\) | \(207530301091125281552569/805586668007040\) | \(431699385371892551884890000000\) | \([2]\) | \(619315200\) | \(4.8997\) |
Rank
sage: E.rank()
The elliptic curves in class 162450n have rank \(0\).
Complex multiplication
The elliptic curves in class 162450n do not have complex multiplication.Modular form 162450.2.a.n
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.