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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 49728.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49728.p1 | 49728bd2 | \([0, -1, 0, -152769, -22931775]\) | \(1504154129818033/5519808\) | \(1446984548352\) | \([2]\) | \(184320\) | \(1.5498\) | |
49728.p2 | 49728bd1 | \([0, -1, 0, -9409, -366911]\) | \(-351447414193/22278144\) | \(-5840081780736\) | \([2]\) | \(92160\) | \(1.2032\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 49728.p have rank \(1\).
Complex multiplication
The elliptic curves in class 49728.p do not have complex multiplication.Modular form 49728.2.a.p
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.