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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 66654by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66654.bl2 | 66654by1 | \([1, -1, 1, 1587694, 1779129105]\) | \(4101378352343/15049939968\) | \(-1624161875825883921408\) | \([2]\) | \(4055040\) | \(2.7528\) | \(\Gamma_0(N)\)-optimal |
66654.bl1 | 66654by2 | \([1, -1, 1, -15932786, 21388050321]\) | \(4144806984356137/568114785504\) | \(61309904070748006777824\) | \([2]\) | \(8110080\) | \(3.0994\) |
Rank
sage: E.rank()
The elliptic curves in class 66654by have rank \(1\).
Complex multiplication
The elliptic curves in class 66654by do not have complex multiplication.Modular form 66654.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.