Properties

Label 259920.fk
Number of curves $2$
Conductor $259920$
CM no
Rank $1$
Graph

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Show commands: 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)
 
\(q + q^{5} + q^{7} - 3 q^{11} - 2 q^{13} + 6 q^{17} + O(q^{20})\) Copy content Toggle raw display

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.