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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 169065.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169065.r1 | 169065m2 | \([0, 0, 1, -11172162, 11944359090]\) | \(30326094659584/5430160125\) | \(27614140845489085051125\) | \([3]\) | \(12690432\) | \(3.0252\) | |
169065.r2 | 169065m1 | \([0, 0, 1, -3213102, -2215206603]\) | \(721403674624/616005\) | \(3132586966121096445\) | \([]\) | \(4230144\) | \(2.4759\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 169065.r have rank \(1\).
Complex multiplication
The elliptic curves in class 169065.r do not have complex multiplication.Modular form 169065.2.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.