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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 169650.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169650.r1 | 169650fe2 | \([1, -1, 0, -2567067, -1582444909]\) | \(-6083088015781323/11781250\) | \(-3623286621093750\) | \([]\) | \(3234816\) | \(2.2390\) | |
169650.r2 | 169650fe1 | \([1, -1, 0, -22317, -3474659]\) | \(-2913790403187/10716526600\) | \(-4521034659375000\) | \([]\) | \(1078272\) | \(1.6897\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 169650.r have rank \(0\).
Complex multiplication
The elliptic curves in class 169650.r do not have complex multiplication.Modular form 169650.2.a.r
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.