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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 90009a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
90009.a2 | 90009a1 | \([1, -1, 1, -13651610, -19410984160]\) | \(385964865476398673601625/387990725193\) | \(282845238665697\) | \([2]\) | \(1320960\) | \(2.4968\) | \(\Gamma_0(N)\)-optimal |
90009.a1 | 90009a2 | \([1, -1, 1, -13654895, -19401172522]\) | \(386243557820661661107625/386967157825467897\) | \(282099058054766096913\) | \([2]\) | \(2641920\) | \(2.8434\) |
Rank
sage: E.rank()
The elliptic curves in class 90009a have rank \(1\).
Complex multiplication
The elliptic curves in class 90009a do not have complex multiplication.Modular form 90009.2.a.a
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.