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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 259920.fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259920.fk1 | 259920fk2 | \([0, 0, 0, -1661201787, 26089006081066]\) | \(-27692833539889/35156250\) | \(-643614811478917776000000000\) | \([]\) | \(141834240\) | \(4.0524\) | |
259920.fk2 | 259920fk1 | \([0, 0, 0, 27758373, 166507753354]\) | \(129205871/729000\) | \(-13345996730826839003136000\) | \([]\) | \(47278080\) | \(3.5031\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 259920.fk have rank \(1\).
Complex multiplication
The elliptic curves in class 259920.fk do not have complex multiplication.Modular form 259920.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.