Properties

Label 38640.cz
Number of curves $2$
Conductor $38640$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cz1")
 
E.isogeny_class()
 

Elliptic curves in class 38640.cz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.cz1 38640da1 \([0, 1, 0, -82640, 9109908]\) \(15238420194810961/12619514880\) \(51689532948480\) \([2]\) \(161280\) \(1.5597\) \(\Gamma_0(N)\)-optimal
38640.cz2 38640da2 \([0, 1, 0, -64720, 13188500]\) \(-7319577278195281/14169067365600\) \(-58036499929497600\) \([2]\) \(322560\) \(1.9063\)  

Rank

sage: E.rank()
 

The elliptic curves in class 38640.cz have rank \(1\).

Complex multiplication

The elliptic curves in class 38640.cz do not have complex multiplication.

Modular form 38640.2.a.cz

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{7} + q^{9} + 2 q^{13} + q^{15} - 4 q^{17} - 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.