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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 133952be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
133952.cg2 | 133952be1 | \([0, -1, 0, -29825, 2352161]\) | \(-11192824869409/2563305472\) | \(-671955149651968\) | \([2]\) | \(958464\) | \(1.5648\) | \(\Gamma_0(N)\)-optimal |
133952.cg1 | 133952be2 | \([0, -1, 0, -500865, 136598561]\) | \(53008645999484449/2060047808\) | \(540029172580352\) | \([2]\) | \(1916928\) | \(1.9114\) |
Rank
sage: E.rank()
The elliptic curves in class 133952be have rank \(1\).
Complex multiplication
The elliptic curves in class 133952be do not have complex multiplication.Modular form 133952.2.a.be
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.