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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 489762.fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.fk1 | 489762fk2 | \([1, -1, 1, -11903378, 15809589969]\) | \(53008645999484449/2060047808\) | \(7248780371761725888\) | \([2]\) | \(40255488\) | \(2.7035\) | \(\Gamma_0(N)\)-optimal* |
489762.fk2 | 489762fk1 | \([1, -1, 1, -708818, 271540689]\) | \(-11192824869409/2563305472\) | \(-9019615137137160192\) | \([2]\) | \(20127744\) | \(2.3569\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 489762.fk have rank \(0\).
Complex multiplication
The elliptic curves in class 489762.fk do not have complex multiplication.Modular form 489762.2.a.fk
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 LMFDB 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 LMFDB labels.