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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 50715y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
50715.x1 | 50715y1 | \([0, 0, 1, -9408, 1086538]\) | \(-1073741824/5325075\) | \(-456711026784075\) | \([]\) | \(165888\) | \(1.4974\) | \(\Gamma_0(N)\)-optimal |
50715.x2 | 50715y2 | \([0, 0, 1, 83202, -26594591]\) | \(742692847616/3992296875\) | \(-342403816849171875\) | \([]\) | \(497664\) | \(2.0467\) |
Rank
sage: E.rank()
The elliptic curves in class 50715y have rank \(2\).
Complex multiplication
The elliptic curves in class 50715y do not have complex multiplication.Modular form 50715.2.a.y
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.