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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 63504by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
63504.cp1 | 63504by1 | \([0, 0, 0, -4851, 142002]\) | \(-35937/4\) | \(-1405192126464\) | \([]\) | \(82944\) | \(1.0683\) | \(\Gamma_0(N)\)-optimal |
63504.cp2 | 63504by2 | \([0, 0, 0, 30429, -203742]\) | \(109503/64\) | \(-1821128995897344\) | \([]\) | \(248832\) | \(1.6176\) |
Rank
sage: E.rank()
The elliptic curves in class 63504by have rank \(1\).
Complex multiplication
The elliptic curves in class 63504by do not have complex multiplication.Modular form 63504.2.a.by
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.