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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 142912bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
142912.e1 | 142912bf1 | \([0, 1, 0, -4669, 121251]\) | \(10994826754048/171941\) | \(176067584\) | \([2]\) | \(110592\) | \(0.71660\) | \(\Gamma_0(N)\)-optimal |
142912.e2 | 142912bf2 | \([0, 1, 0, -4529, 129007]\) | \(-627200828368/86191567\) | \(-1412162633728\) | \([2]\) | \(221184\) | \(1.0632\) |
Rank
sage: E.rank()
The elliptic curves in class 142912bf have rank \(2\).
Complex multiplication
The elliptic curves in class 142912bf do not have complex multiplication.Modular form 142912.2.a.bf
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.