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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 16576.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16576.p1 | 16576q1 | \([0, 0, 0, -67528, 6754200]\) | \(33256413948450816/2481997\) | \(2541564928\) | \([2]\) | \(39168\) | \(1.2543\) | \(\Gamma_0(N)\)-optimal |
16576.p2 | 16576q2 | \([0, 0, 0, -67388, 6783600]\) | \(-2065624967846736/17960084863\) | \(-294258030395392\) | \([2]\) | \(78336\) | \(1.6009\) |
Rank
sage: E.rank()
The elliptic curves in class 16576.p have rank \(0\).
Complex multiplication
The elliptic curves in class 16576.p do not have complex multiplication.Modular form 16576.2.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.