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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 145728bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
145728.ej2 | 145728bt1 | \([0, 0, 0, -3360, -72808]\) | \(5619712000/184437\) | \(137681482752\) | \([2]\) | \(147456\) | \(0.91072\) | \(\Gamma_0(N)\)-optimal |
145728.ej1 | 145728bt2 | \([0, 0, 0, -8220, 185744]\) | \(5142706000/1728243\) | \(20642023784448\) | \([2]\) | \(294912\) | \(1.2573\) |
Rank
sage: E.rank()
The elliptic curves in class 145728bt have rank \(1\).
Complex multiplication
The elliptic curves in class 145728bt do not have complex multiplication.Modular form 145728.2.a.bt
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.